% -*- coding: utf-8; time-stamp-format: "%02d-%02m-%:y at %02H:%02M:%02S %Z" ; time-stamp-active: nil -*-
%<*dtx>
\def\zeckdtxtimestampaux #1<#2>{#2}
\edef\zeckdtxtimestamp{\zeckdtxtimestampaux Time-stamp: <06-10-2025 at 20:57:57 CEST>}
%</dtx>
%<*drv>
%% ---------------------------------------------------------------
\def\zeckpackdate{2025/10/06}%
\def\zeckversion {0.9alpha}%
%</drv>
%<*dtx>
\catcode0 0 \catcode`\\ 12
^^@iffalse
%</dtx>
%<*readme>--------------------------------------------------------
| Source:   zeckendorf.dtx
| Version:  0.9alpha, 2025/10/06
| Author:   Jean-François Burnol
| Keywords: Zeckendorf representation; Knuth multiplication
| License:  LPPL 1.3c

Aim
===

This package adds to `\xintiieval`'s syntax the `$` character as
infix operator computing the Knuth multiplication of positive
integers.  It adds `fib()` and `fibseq()` functions to compute
Fibonacci numbers and `zeck()`, `zeckindex()` to compute the
Zeckendorf indices of positive integers.

Conversion between integers and Zeckendorf binary words (or lists
of indices as done by `zeck()` inside `\xintiieval{}`) is
available through macros.  Of course they all work using expansion
only, and as the package is an add-on to `xintexpr` it handles
arbitrarily big integers (up to thousands of digits at most).

It can be used in three ways:
- as a LaTeX package: `\usepackage{zeckendorf}`,
- as a Plain e-TeX macro file: `\input zeckendorfcore.tex`,
- or interactively on the command line: `etex zeckendorf`. If
  available use rather `rlwrap etex zeckendorf` for a smoother
  experience.

The package is in alpha stage and its user interface may change
in backwards incompatible ways.
Of course, suggestions are most welcome.

Installation
============

Exert `etex zeckendorf.dtx` to extract macro files.  To build the
documentation run `xelatex` on the extracted `zeckendorf-doc.tex`,
or directly on `zeckendorf.dtx`.

The recommended TDS structure is:

    /tex/generic/zeckendorf/zeckendorfcore.tex
    /tex/generic/zeckendorf/zeckendorf.tex
    /tex/generic/zeckendorf/zeckendorf.sty
    /source/generic/zeckendorf/zeckendorf.dtx
    /doc/generic/zeckendorf/zeckendorf-doc.pdf
    /doc/generic/zeckendorf/README.md

License
=======

Copyright © 2025 Jean-François Burnol

| This Work may be distributed and/or modified under the
| conditions of the LaTeX Project Public License 1.3c.
| This version of this license is in

>   <http://www.latex-project.org/lppl/lppl-1-3c.txt>

| and version 1.3 or later is part of all distributions of
| LaTeX version 2005/12/01 or later.

This Work has the LPPL maintenance status "author-maintained".

The Author and Maintainer of this Work is Jean-François Burnol.

This Work consists of the main source file and its derived files

    zeckendorf.dtx,
    zeckendorfcore.tex, zeckendorf.tex, zeckendorf.sty,
    README.md, zeckendorf-doc.tex, zeckendorf-doc.pdf

%</readme>--------------------------------------------------------
%<*!readme>
%% ---------------------------------------------------------------
%% The zeckendorf package: Zeckendorf representation of big integers
%% Copyright (C) 2025 Jean-Francois Burnol
%%
%</!readme>
%<*changes>-------------------------------------------------------
\item[0.9alpha (2025/10/06)] Initial release.
%</changes>-------------------------------------------------------
%<*drv>-----------------------------------------------------------
%% latex zeckendorf-doc.tex (thrice) && dvipdfmx zeckendorf-doc.dvi
%% to produce zeckendorf-doc.pdf
%%
%% Doing latexmk -pdf zeckendorf-doc.tex is also possible but
%% will produce a bigger file due usage of pdflatex.
%%
\NeedsTeXFormat{LaTeX2e}[2020/02/02]
\ProvidesFile{zeckendorf-doc.tex}%
[\zeckpackdate\space v\zeckversion\space driver file for %
  zeckendorf documentation (JFB)]%
\PassOptionsToClass{a4paper,fontsize=11pt}{scrartcl}
\RequirePackage{iftex}
\chardef\Withdvipdfmx \iftutex0 \else\ifpdf0 \else1 \fi\fi
\chardef\NoSourceCode 0 % replace 0 by 1 for not including source code
\input zeckendorf.dtx
%%% Local Variables:
%%% mode: latex
%%% End:
%</drv>-----------------------------------------------------------
%<*dtx>
^^@fi ^^@catcode`\ 0 \catcode0 12
\def\hiddenifs{\iftutex0 \else\ifpdf0 \else1 \fi\fi}%
\chardef\noetex 0
\ifx\numexpr\undefined \chardef\noetex 1 \fi
\ifnum\noetex=1 \chardef\extractfiles 0 % extract files, then stop
\else
    \ifx\ProvidesFile\undefined
      \chardef\extractfiles 0 % no LaTeX2e; etex, ... on zeckendorf.dtx
    \else % latex/pdflatex on zeckendorf-doc.tex or on zeckendorf.dtx
      \NeedsTeXFormat{LaTeX2e}[2020/02/02]%
      \ProvidesFile{zeckendorf.dtx}[bundle source (\zeckpackdate)]%
      \ifx\Withdvipdfmx\undefined
        % latex run is on zeckendorf.dtx, we will extract all files
        \chardef\extractfiles 1 % 1 = extract and typeset, 2=only typeset
        \chardef\NoSourceCode 0 % 0 = include source code, 1 = do not
        \PassOptionsToClass{a4paper,fontsize=11pt}{scrartcl}%
        \RequirePackage{iftex}\relax
        \chardef\Withdvipdfmx \hiddenifs
      \else % latex run is on zeckendorf-doc.tex,
        \chardef\extractfiles 2 % no extractions
      \fi
    \fi
\fi
\ifnum\extractfiles<2 % we extract files
 \def\MessageDeFin{\newlinechar10 \let\Msg\message
 \Msg{^^J}%
 \Msg{********************************************************************^^J}%
 \Msg{*^^J}%
 \Msg{* To finish the installation you have to move the following^^J}%
 \Msg{* files into a directory searched by TeX:^^J}%
 \Msg{*^^J}%
 \Msg{*\space\space\space\space zeckendorf.sty^^J}%
 \Msg{*\space\space\space\space zeckendorf.tex^^J}%
 \Msg{*\space\space\space\space zeckendorfcore.tex^^J}%
 \Msg{*^^J}%
 \Msg{* Happy TeXing!^^J}%
 \Msg{*^^J}%
 }%
 \ifnum\extractfiles=0 % only extraction
  \expandafter\def\expandafter\MessageDeFin\expandafter{%
  \MessageDeFin
  \Msg{* To produce the documentation run latex thrice on zeckendorf-doc.tex^^J}%
  \Msg{* then dvipdfmx on zeckendorf-doc.dvi.^^J}%
  \Msg{*^^J}%
  \Msg{********************************************************************^^J}%
  }%
 \else %% *latex run on zeckendorf.dtx
  \ifnum\Withdvipdfmx=1 % for dvipdfmx
   \expandafter\def\expandafter\MessageDeFin\expandafter{%
   \MessageDeFin
   \Msg{* Please use dvipdfmx once the LaTeX runs are done, and^^J}%
   \Msg{* consider renaming zeckendorf.pdf to zeckendorf-doc.pdf^^J}%
   \Msg{********************************************************************^^J}%
   }%
  \else % pdflatex, lualatex or xelatex
   \expandafter\def\expandafter\MessageDeFin\expandafter{%
   \MessageDeFin
   \Msg{* Please consider renaming zeckendorf.pdf to zeckendorf-doc.pdf^^J}%
   \Msg{********************************************************************^^J}%
   }%
  \fi
 \fi
 % file extraction
 \begingroup
    \input docstrip.tex
    \askforoverwritefalse
    \generate{\nopreamble\nopostamble
    \file{README.md}{\from{zeckendorf.dtx}{readme}}
    \usepreamble\defaultpreamble
    \usepostamble\defaultpostamble
    \file{zeckendorfchanges.tex}{\from{zeckendorf.dtx}{changes}}
    \file{zeckendorf-doc.tex}{\from{zeckendorf.dtx}{drv}}
    \file{zeckendorfcore.tex}{\from{zeckendorf.dtx}{core}}
    \file{zeckendorf.tex}{\from{zeckendorf.dtx}{interactive}}
    \file{zeckendorf.sty}{\from{zeckendorf.dtx}{sty}}}
 \endgroup
\else % *latex on zeckendorf-doc.tex we skipped extraction
 \ifnum\Withdvipdfmx=1
  \def\MessageDeFin{\newlinechar10 \let\Msg\message
  \Msg{^^J}% too lazy to add the multipls spaces to make a frame
  \Msg{********************************************************************^^J}%
  \Msg{* Please use dvipdfmx once the LaTeX runs are done^^J}%
  \Msg{********************************************************************^^J}%
  }%
 \fi
\fi
\ifnum\extractfiles=0
% direct tex/etex/xetex on zeckendorf.dtx, files now extracted, stop
  \MessageDeFin\expandafter\end
\fi
\ifdefined\MessageDeFin\AtEndDocument{\MessageDeFin}\fi
%-------------------------------------------------------------------------------
\documentclass {scrartcl}
%%% START OF CUSTOM doc.sty LOADING (May 21, 2022)
\usepackage{doc}[=v2]
% As explained above I was formerly using scrdoc hence ltxdoc indirectly.
% Let's emulate here the little I appear to need from ltxdoc.cls and
% srcdoc.cls.
%
\AtBeginDocument{\MakeShortVerb{\|}}
\DeclareFontShape{OT1}{cmtt}{bx}{n}{<-> ssub * cmtt/m/n}{}
\DeclareFontFamily{OMS}{cmtt}{\skewchar\font 48}  % '60
\DeclareFontShape{OMS}{cmtt}{m}{n}{<-> ssub * cmsy/m/n}{}
\DeclareFontShape{OMS}{cmtt}{bx}{n}{<-> ssub * cmsy/b/n}{}
\DeclareFontShape{OT1}{cmss}{m}{it}{<->ssub*cmss/m/sl}{}
\ifnum\NoSourceCode=1
  \OnlyDescription
\fi
\CodelineNumbered
\DisableCrossrefs
% \setcounter{StandardModuleDepth}{1} % we don't use this mechanism currently
\def\cmd#1{\cs{\expandafter\cmd@to@cs\string#1}}
\def\cmd@to@cs#1#2{\char\number`#2\relax}
% This is hyperref compatible (contrarily to using \bslash)
% I need the \detokenize due to usage in implementation part with names
% containing the underscore for example
\DeclareRobustCommand\cs[1]{\texttt{\textbackslash\detokenize{#1}}}
\providecommand\marg[1]{%
  {\ttfamily\char`\{}\meta{#1}{\ttfamily\char`\}}}
\providecommand\oarg[1]{%
  {\ttfamily[}\meta{#1}{\ttfamily]}}
\providecommand\parg[1]{%
  {\ttfamily(}\meta{#1}{\ttfamily)}}

% There is very little we seem to need from the scrdoc extras: page geometry
% is set by geometry package and a4paper option from zeckendorf.tex file.  So it
% seems I only need the hologo loading:
\usepackage{hologo}
\DeclareRobustCommand*{\eTeX}{\hologo{eTeX}}%
%\DeclareRobustCommand*{\LuaTeX}{\hologo{LuaTeX}}%
%
%%% end of ltxdoc+srcdoc emulation.

\ifnum\NoSourceCode=1 \OnlyDescription\fi
\usepackage{ifpdf}
\ifpdf\chardef\Withdvipdfmx 0 \fi
\makeatletter
\ifnum\Withdvipdfmx=1
% I think those are mostly obsolete...
   \@for\@tempa:=hyperref,bookmark,graphicx,xcolor,pict2e\do
            {\PassOptionsToPackage{dvipdfmx}\@tempa}
   %
   \PassOptionsToPackage{dvipdfm}{geometry}
   \PassOptionsToPackage{bookmarks=true}{hyperref}
   \PassOptionsToPackage{dvipdfmx-outline-open}{hyperref}
   %\PassOptionsToPackage{dvipdfmx-outline-open}{bookmark}
   %
   \def\pgfsysdriver{pgfsys-dvipdfm.def}
\else
   \PassOptionsToPackage{bookmarks=true}{hyperref}
\fi
\makeatother

\pagestyle{headings}

\iftutex\else % but newtxtt requires it anyhow
\usepackage[T1]{fontenc}
\fi

\usepackage[hscale=0.66,vscale=0.75]{geometry}

\usepackage{amsmath}
\allowdisplaybreaks

% requires newtxtt 1.05 or later
\usepackage[zerostyle=d,scaled=0.95,straightquotes]{newtxtt}
\renewcommand\familydefault\ttdefault
\usepackage[noendash]{mathastext}
\renewcommand\familydefault\sfdefault

\usepackage{graphicx}

\usepackage[dvipsnames,svgnames]{xcolor}

\definecolor{joli}{RGB}{225,95,0}
\definecolor{JOLI}{RGB}{225,95,0}
\definecolor{BLUE}{RGB}{0,0,255}
%% \definecolor{niceone}{RGB}{38,128,192}% utilisé avant pour urlcolor
%% \colorlet{jfverbcolor}{yellow!5}
\colorlet{shortverbcolor}{magenta}% 1.6 release NavyBlue RoyalPurple
\colorlet{softwrapcolor}{blue}
\colorlet{digitscolor}{olive}% 1.6 {OrangeRed}

% transféré de xint-manual.tex (maintenant dans xint.dtx)
\DeclareFontFamily{U}{MdSymbolC}{}
\DeclareFontShape {U}{MdSymbolC}{m}{n}{<-> MdSymbolC-Regular}{}

\makeatletter
\newbox\cdbx@SoftWrapIcon
\def\cdbx@SetSoftWrapBox{%
    \setbox\cdbx@SoftWrapIcon\hb@xt@\z@
    {\hb@xt@\fontdimen2\font
        {\hss{\color{softwrapcolor}\usefont{U}{MdSymbolC}{m}{n}\char"97}\hss}%
     \hss}%
   }
\makeatother

\usepackage[english]{babel}

\usepackage{hyperref}
\hypersetup{%
%linktoc=all,%
breaklinks=true,%
colorlinks=true,%
urlcolor=Cerulean,% 1.6 ProcessBlue,%SkyBlue
linkcolor=PineGreen,%
pdfauthor={Jean-François Burnol},%
pdftitle={The Zeckendorf package},%
pdfsubject={Arithmetic with TeX},%
pdfkeywords={Expansion, arithmetic, TeX},%
pdfstartview=FitH,%
pdfpagemode=UseOutlines}
\usepackage{bookmark}

% importé de xint 2021/05/14
\newcommand\RaisedLabel[2][6]{%
\vspace*{-#1\baselineskip}%
\begingroup
  \let\leavevmode\relax\phantomsection
  \label{#2}%
\endgroup
\vspace*{#1\baselineskip}%
}

%---- \verb, and verbatim like `environments'. \MicroFont et \MacroFont
\makeatletter
\def\MicroFont {\ttfamily\cdbx@SetSoftWrapBox\color{shortverbcolor}}
%\def\MicroFont {\ttfamily \color[named]{OrangeRed}\cdbx@SetSoftWrapBox }

% \MacroFont est utilisé par macrocode, mais sa définition est figée dans
% \macro@font au \begin{document}

\def\MacroFont {\ttfamily \baselineskip12pt\relax }

%--- November 4, 2014 making quotes straight. (maintenu pour *)
%
% there is no hook in \macrocode after \dospecials etc. Thus I will need to
% take the risk that some future evolution of doc.sty (or perhaps scrdoc)
% invalidates the following.
%
% Actually, I should not at all rely on the doc class, I should do it all by
% myself.

\def\macrocode{\macro@code
               \frenchspacing \@vobeyspaces
               \makestarlowast % \makequotesstraight (obsolète)
               \xmacro@code }

%--- lower * symbol in text

\def\lowast{\raisebox{-.25\height}{*}}
\catcode`* \active
\def\makestarlowast {\let*\lowast\catcode`\*\active}%
\catcode`* 12

\def\verb #1%
{%
  \relax\leavevmode\null
  \begingroup
   \MicroFont
   \def\@makeletter##1{\catcode`##1=11\relax}%
   % \scantokens will have a result of inserting a space after cs's.
   % hence the need to have the catcodes of things like _ right.
   % I also need < for > for code comments
   % \dospecials at begin document
   % \do\ \do\\\do\{\do\}\do\$\do\&\do\#\do\^\do\_\do\%\do\~\do\|
   \odef\dospecials{\dospecials\do\:\do\<\do\>\do\-\do\+}%
   % naturally won't work in footnotes though.
   % this code is truly not satisfying, but enough for my needs here.
   \catcode`\\ 11 \catcode`## 11
   % I don't do \catcode`\% 11 to avoid surprises in code comments
   % if my |...| has a linebreak
   \def\@jfverb ##1#1{\let\do\@makeletter \dospecials
   % lowering a bit the *
                      \makestarlowast
   %\let\do\do@noligs  \verbatim@nolig@list % not needed here
                      \@vobeyspaces\everyeof{\noexpand}%
                      \expandafter\@@jfverb\scantokens{##1}\relax}%
  \@jfverb
}%

\def\@@jfverb #1{\ifcat\noexpand#1\relax\endgroup\@\else
       \nobreak\hskip0pt plus 1pt % added 2025/10/06 for Zeckendorf
       \discretionary{\copy\cdbx@SoftWrapIcon}{}{}%
       #1\expandafter\@@jfverb\fi }

\makeatother


% everbatim environment
% =====================

% October 13-14, 2014
% Verbatim with an \everypar hook, mainly to have background color, followed by
% execution of the contents (not limited by a group-scope)

\makeatletter
\colorlet{everbatimfgcolor}{DarkBlue}
\colorlet{everbatimbgcolor}{Beige}
\colorlet{everbatimxfgcolor}{Maroon}

\def\everbatim {\s@everbatim\@everbatim}
\@namedef{everbatim*}{\s@everbatim\@everbatimx}

% Note: one can not use everbatim inside itself or everbatim* inside itself
\def\s@everbatim {%
     \everbatimtop % put there size changes
       \topsep    \z@skip
       \partopsep \z@skip
       \itemsep   \z@skip
       \parsep    \z@skip
       \parskip   \z@skip
       \lineskip  \z@skip
     \let\do\@makeother \dospecials
     \let\do\do@noligs  \verbatim@nolig@list
     \makestarlowast
     \everbatimhook
     \trivlist
       \@topsepadd \z@skip
       \item\relax
       \leftskip   \@totalleftmargin
       \rightskip  \z@skip
       \parindent  \z@
       \parfillskip\@flushglue
       \parskip    \z@skip
       \@@par
       \def\par{\leavevmode\null\@@par\pagebreak[1]}%
       \everypar\expandafter{\the\everypar \unpenalty
                \everbatimeverypar
                \everypar \expandafter{\the\everypar\everbatimeverypar}%
       }%
       \obeylines \@vobeyspaces
}
% 27 mai 2022, plus de \small
\def\everbatimtop {\MacroFont }%
\let\everbatimhook\empty
\def\everbatimeverypar{\strut
                   {\color{everbatimbgcolor}\vrule\@width\linewidth }%
                   \kern-\linewidth
                   \kern\everbatimindent }
\def\everbatimindent {\z@}% voir plus loin atbegindocument

\begingroup
\lccode`X 13
\catcode`X \active
\lccode`Y `* % this is because of \makestarlowast.
% I have to think whether this is useful: obviously if I were to provide
% everbatim and everbatim*  in a package I wouldn't do that.
\catcode`Y  \active
\catcode`| 0 \catcode`[ 1 \catcode`] 2 \catcode`* 12
\catcode`{ 12 \catcode`} 12 |catcode`\\ 12
|lowercase[|endgroup% both freezes catcodes and converts X to active ^^M
|def|@everbatim #1X#2\end{everbatim}%
  [#2|end[everbatim]|everbatimbottom ]
|def|@everbatimx #1X#2\end{everbatimY}]%
  {#2\end{everbatim*}%
% No group here: this allows executed code to make macro
% definitions which may reused in later uses of everbatim.
% refactored 2022/01/11, rather than passing \newlinechar value
% as was done formerly via everbatim* (see above) and fetching it here as #1 
% it is thus assumed executed contents do not terminate a scope
     \edef\everbatimrestorenewlinechar{\newlinechar\the\newlinechar\relax}%
     \newlinechar 13
% refactored 2022/01/11 to fix a \parskip issue
% attention, \parskip thus set to zero for execution of contents
% reason: avoid extra space if everbatim* is in an \item of a list
% between verbatim and output of execution, if it starts a paragraph
% a \vskip-\parskip approach (cf former \everbatimundoparskip)
% would be no good in case contents create a display
     \edef\everbatimrestoreparskip{\parskip\the\parskip\relax}%
     \parskip\z@skip
% execution of the contents (expected to be LaTeX code...)
     \everbatimxprehook
     \scantokens {#2}%
     \everbatimrestorenewlinechar
     \everbatimrestoreparskip
     \everbatimxposthook
% input after \end{everbatim*} on same line in source is allowed
}%
% L'espace venant du endofline final mis par \scantokens sera inhibé si #2 se
% termine par un % ou un \x, etc...
\let\everbatimbottom\empty

\def\endeverbatim{\if@newlist \leavevmode\fi\endtrivlist}
\@namedef{endeverbatim*}{\endeverbatim}
\def\everbatimxprehook {\colorlet{everbsavedcolor}{.}%
                        \color{everbatimxfgcolor}}%
\def\everbatimxposthook{\color{everbsavedcolor}}
{\sbox0{\color{everbatimxfgcolor}\xdef\@tempa{\current@color}}}
\ifpdf
 \ifluatex
   \edef\everbatimxprehook
       {\pdfextension colorstack\noexpand\@pdfcolorstack push{\@tempa}\relax}
   \def\everbatimxposthook
       {\pdfextension colorstack\@pdfcolorstack pop\relax}
 \else
   \edef\everbatimxprehook
      {\pdfcolorstack\noexpand\@pdfcolorstack push{\@tempa}\relax}
   \def\everbatimxposthook{\pdfcolorstack\@pdfcolorstack pop\relax}
 \fi
\else
  \ifxetex
   \edef\everbatimxprehook{\special{color push \@tempa}}
   \def\everbatimxposthook{\special{color pop}}
  \else
    \ifnum\Withdvipdfmx=1
     \edef\everbatimxprehook{\special{color push \@tempa}}
     \def\everbatimxposthook{\special{color pop}}
    \fi
  \fi
\fi
\makeatother

% 2025/10/05
% Toujours pénible de devoir se battre avec les espaces verticaux de LaTeX.
% J'ai dû réexaminer la situation pour Zeckendorf, différent de bnumexpr
% où j'utilise \[..\] pour les \bneshoweval etc...
\AtBeginDocument{%
% \message{ICI below: \the\belowdisplayskip^^J%
%              belowshort: \the\belowdisplayshortskip^^J%
%              above: \the\abovedisplayskip^^J%
%              aboveshor: \the\abovedisplayshortskip^^J%
% }%
% below: 11.0pt plus 3.0pt minus 6.0pt
% belowshort: 6.5pt plus 3.5pt minus 3.0pt
% above: 11.0pt plus 3.0pt minus 6.0pt
% aboveshor: 0.0pt plus 3.0pt
   \belowdisplayskip\belowdisplayshortskip
   \abovedisplayskip\belowdisplayskip
}

\usepackage{xspace}

\def\zeckname
   {\href{http://www.ctan.org/pkg/zeckendorf}{zeckendorf}\xspace }%

\def\zecknameimp
   {\texorpdfstring
          {\hyperref[part:zeckendorfcode]{{\color{shortverbcolor}\ttzfamily zeckendorf}}}
          {zeckendorf}%
    \xspace }%

\def\zecknameuserdoc
   {\texorpdfstring
          {\hyperref[titlepage]{{\color{PineGreen}\ttzfamily zeckendorf}}}
          {zeckendorf}%
    \xspace }%

\def\bnumexprname
   {\href{http://www.ctan.org/pkg/bnumexpr}{bnumexpr}\xspace }%
\def\xintkernelname
   {\href{http://www.ctan.org/pkg/xint}{xintkernel}\xspace }%
\def\xintcorename
   {\href{http://www.ctan.org/pkg/xint}{xintcore}\xspace }%
\def\xintname
   {\href{http://www.ctan.org/pkg/xint}{xint}\xspace }%
\def\xinttoolsname
   {\href{http://www.ctan.org/pkg/xint}{xinttools}\xspace }%
\def\xintfracname
   {\href{http://www.ctan.org/pkg/xint}{xintfrac}\xspace }%
\def\xintexprname
   {\href{http://www.ctan.org/pkg/xintexpr}{xintexpr}\xspace }%
\def\xintbinhexname
   {\href{http://www.ctan.org/pkg/xint}{xintbinhex}\xspace }%

\DeclareRobustCommand\ctanpackage[1]{\href{https://ctan.org/pkg/#1}{#1}}

\usepackage{zeckendorf}

\usepackage{etoc}
\usepackage{centeredline}
\usepackage{framed}
\makeatletter\let\check@percent\relax\makeatother
\usepackage[autolanguage,np]{numprint}
\usepackage{enumitem}

\newenvironment{TeXnote}{\par\footnotesize\medskip\noindent\textbf{\TeX-nical note:
  }}{\par\medskip}

\newcommand\dgts{\textcolor{digitscolor}}
\newcommand\mdgts{\mathcolor{digitscolor}}

\begin{document}
\thispagestyle{empty}
\ttzfamily
\pdfbookmark[1]{Title page}{TOP}

{%
\normalfont\Large\parindent0pt \parfillskip 0pt\relax
 \leftskip 2cm plus 1fil \rightskip 2cm plus 1fil
 The \zeckname package\par
}

\RaisedLabel{titlepage}

{\centering
  \textsc{Jean-François Burnol}\par
  \footnotesize
  jfbu (at) free (dot) fr\par
  Package version: \zeckversion\ (\zeckpackdate)\par
  {From source file \texttt{zeckendorf.dtx} of \texttt{\zeckdtxtimestamp}}\par
}

\etocsetnexttocdepth{subsection}
\etocsettagdepth{code}{section}
\etocsettocstyle{}{}
\etocdefaultlines
\tableofcontents

\part{User manual}
\section{Mathematical background}
\label{sec:math}

Let us recall that the Fibonacci sequence starts with $F_0=0$,
$F_1=1$, and obeys the recurrence $F_n=F_{n-1} + F_{n-2}$ for $n\geq2$.  So $F_2=1$,
$F_3=2$, $F_4=3$ and by a simple induction $F_k = k-1$.  Ahem, not at all!
Here are the first few, starting at $F_2=1$:
\[
\xintiieval{seq(fib(n), n=2..18)}\dots
\]

\def\testN{100000000}
\oodef\testword{\ZeckWord{\testN}}
\oodef\testmaxindex{\ZeckIndex{\testN}}
\ifnum\xintNthElt{0}{\testword}=\numexpr\testmaxindex-1\relax\else\ERROR\fi

\textbf{Zeckendorf}'s Theorem says that any positive integer has a unique
representation as a sum of the Fibonacci numbers $F_n$, $n\geq2$, under the
conditions that no two indices differ by one, and that no index is repeated.
For example:
\begin{align*}
\xintFor #1 in {10,100,1000,10000,100000, 1000000, 10000000}\do{
\np{#1} &= \mdgts{\ZeckExplicitSum{#1}}\\
}
\np{\testN} &= \mdgts{\ZeckIndexedSum{\testN}}
\end{align*}
This is called the Zeckendorf representation, and it can be given either as
above, or as the list of the indices (in decreasing or increasing order), or
as a binary word which in the examples above are
%
\begin{align*}
\xintFor #1 in {10,100,1000,10000,100000, 1000000, 10000000}\do{
\np{#1} &= \mdgts{\ZeckWord{#1}_{\text{Zeckendorf}}}\\
}
\np{\testN} &= \mdgts{\ZeckWord{\testN}_{\text{Zeckendorf}}}\\
\np{1000000000} &= \mdgts{\ZeckWord{1000000000}_{\text{Zeckendorf}}}
\end{align*}
%
The least significant digit says whether the Zeckendorf representation uses
$F_2$ and so on from right to left (one may prefer to put the binary digits in
the reverse order, but doing as above is more reminiscent of binary, decimal,
or other representations using a given radix). In the next-to-last example the word
length is $\testmaxindex-2+1=\mdgts{\the\numexpr\testmaxindex-1\relax}$, and in
general it is $K-1$ where $K$ is the largest index such that $F_K$ is at most
equal to the given positive integer.  For $\np{1000000000}$ this maximal index is
\dgts{\ZeckIndex{1000000000}} and indeed the associated word has length
\dgts{\expanded{\noexpand\xintLength{\ZeckWord{1000000000}}}}.

In a Zeckendorf binary word the sub-word
\dgts{11} never occurs, and this, combined wih the fact that
the leading digit is \dgts{1}, characterizes the Zeckendorf words.


\textbf{Donald Knuth} (whose name may ring some bells to \TeX{} users) has
shown that defining $ a \circ b$ as $\sum_i\sum_j F_{a_i+b_j}$ where the
$a_i$'s and the $b_j$'s are the indices involved in the respective Zeckendorf
representations of $a$ and $b$ is an \emph{associative} operation on positive
integers (it is obviously commutative).

The Fibonacci recurrence can also be prolungated to negative $n$'s, and it
turns out that $F_{-n}=(-1)^{n-1}F_n$.  \textbf{Donald Knuth} has shown that
any relative integer has a unique representation as a sum of these
``NegaFibonacci'' numbers $F_{-n}$, $n\geq1$, again with the condition that no
index is repeated and no two indices differ by one. In the special case of
zero, the representation is an empty sum.  Here is the sequence of these
``NegaFibonacci'' numbers starting at $n=-1$:
% \[
% \xintiieval{seq((-1)^(n-1)*fib(n), n=1..16)}\dots
% \]
% implémentation négatifs a marché du premier coup...
% Pas envie pour le moment d'implémenter fibseq(a,b) avec b<=a
\[
\xintiieval{*reversed(fibseq(-16,-1))}\dots
\]

% à noter que (-1)^{n-1} au lieu de (-1)^(n-1) donne un résultat très
% contreintuitif car xintexpr va retirer les {..} donc c'est comme
% (-1)^n - 1 donc totalement snas rapport... (si l'input est (-1)^{n-1} * x,
% le 1 sera absorbé par le *x d'ailleurs).

\section{Use on the command line}
\label{sec:interactive}

Open a command line window and execute:
\centeredline{|etex zeckendorf|}
then follow the displayed instructions.

The (\TeX{} Live) |*tex| executables are not linked with the |readline|
library, and this makes interactive use quite painful.  If you are on a decent
system, launch the interactive session rather via
\centeredline{|rlwrap etex zeckendorf|}
for a smoother experience. 


\section{Use as a \LaTeX{} package}
\label{sec:latex}

As expected, add to the preamble: \centeredline{|\usepackage{zeckendorf}|}
There are no options.

\xintexprname is loaded, macros are defined to go from integers to Zeckendorf
representations and back, and to compute the Knuth multiplication of positive
integers.

|\xintiieval| is extended with the functions |fib()|, |fibseq()|, |zeckindex()|
and |zeck()|.  The |$| is added to the syntax as infix operator
(with same precedence as multiplication) doing
the Knuth multiplication.%$ que Emacs/AucTeX est bête

\begin{description}
\item[\cs{ZeckTheFN}]  This macro computes Fibonacci numbers.
\begin{everbatim*}
\ZeckTheFN{100}, \ZeckTheFN{100 + 15}\newline
\end{everbatim*}%
  As shown, the argument can be an integer expression (only in the sense of
  |\inteval|, not in the one of |\xinteval|, for example you can not have
  powers only additions and multiplications).  Negative arguments are allowed:
\begin{everbatim*}
\ZeckTheFN{0}, \ZeckTheFN{-1}, \ZeckTheFN{-2}, \ZeckTheFN{-3},
\ZeckTheFN{-4} 
\end{everbatim*}

The syntax of |\xintiieval| is extended via addition of a |fib()| function, which
gives a convenient interface:
\begin{everbatim*}
\xintiieval{seq(fib(n), n=-5..5, 10, 20, 100)}
\end{everbatim*}
\begin{everbatim*}
\xintiieval{seq(fib(2^n), n=1..7)}
\end{everbatim*}

\item[\cs{ZeckTheFSeq}] This computes not only one but a whole contiguous
  series of Fibonacci numbers but its output format is a sequence of braced
  numbers, and tools such as those of \xinttoolsname are needed to manipulate
  its output.  For this reason it is not further documented here.

  The syntax of |\xintiieval| is extended via addition of a |fibseq()|
  function, which gives a convenient interface:
\begin{everbatim*}
\xintiieval{fibseq(10,20)}\newline
\end{everbatim*}%
  Notice the square brackets used on output.  In the terminology of
  \xintexprname, the function produces a |nutple|.  Use the |*| prefix to
  remove the brackets:
\begin{everbatim*}
\xintiieval{reversed(*fibseq(-20,-10))}
\end{everbatim*}

  \textbf{IMPORTANT:} currently, |fibseq(a,b)| \textbf{falls into an infinite loop} if
  $a\geq b$.  Use it only with $a<b$.  Above we used the |reversed()| function
  to get the output in order from $F_{-10}$ to $F_{-20}$ and not from
  $F_{-20}$ to $F_{-10}$.

\item[\cs{ZeckIndex}] This computes the largest index $k$ such that $F_k\leq x$,
  where $x$ is the input.  The input is only \emph{f}-expanded, if you need it
  to be an expression you must wrap it in |\xintiieval|.

  The syntax of |\xintiieval| is extended via addition of a |zeckindex()|
  function, which gives a more convenient interface.

  \textbf{IMPORTANT:} The input \textbf{must be positive} (for now).  No check
  is made that this is the case.

  \textbf{Note:} Input must not have more than a few thousand decimal digits.

\begin{everbatim*}
\ZeckIndex{123456789123456789123456789123456789}
\end{everbatim*}
\begin{everbatim*}
\ZeckTheFN{\ZeckIndex{123456789123456789123456789123456789}}
\end{everbatim*}
\begin{everbatim*}
\ZeckTheFN{1+\ZeckIndex{123456789123456789123456789123456789}}
\end{everbatim*}
\begin{everbatim*}
\ZeckIndex{\xintiieval{2^100}}
\end{everbatim*}
\begin{everbatim*}
\xintiieval{zeckindex(2^100)}
\end{everbatim*}
\begin{everbatim*}
\xintiieval{fib(zeckindex(2^100))}
\end{everbatim*}
\begin{everbatim*}
\xintiieval{2^100}
\end{everbatim*}
\begin{everbatim*}
\xintiieval{fib(1 + zeckindex(2^100))}
\end{everbatim*}
\begin{everbatim*}
\xintiieval{seq(zeckindex(10^n), n = 0..40)}
\end{everbatim*}

\item[\cs{ZeckIndices}] This computes the Zeck representation as a comma
  separated list of indices.  The input is only \emph{f}-expanded, if you need it
  to be an expression you must wrap it in |\xintiieval|.

  The macro is also known as |\ZeckZeck|.

  The syntax of |\xintiieval| is extended via addition of a |zeck()|
  function, which gives a more convenient interface.

  \textbf{IMPORTANT:} The input \textbf{must be positive}.  No check
  is made that this is the case.

  \textbf{Note:} Input must not have more than a few thousand decimal digits.

\begin{everbatim*}
\ZeckZeck{123456789123456789123456789}
\end{everbatim*}

The |zeck()| function produces a \emph{nutple} (see \xintexprname
documentation), and this shows in the presence of brackets in the output.
You can use the |*| prefix to unpack.
\begin{everbatim*}
\xintiieval{zeck(123456789)}\newline
\xintiieval{reversed(zeck(123456789))}\newline
\xintiieval{*zeck(123456789)}\newline
\xintiieval{*reversed(zeck(123456789))}
\end{everbatim*}

The power of the |\xintiieval| syntax can be demonstrated:
\begin{everbatim*}
\xintiieval{add(fib(n), n = *zeck(123456789))}
\end{everbatim*}
\begin{everbatim*}
\xintiieval{add(fib(n), n = *zeck(123456789123456789123456789))}
\end{everbatim*}

\begin{TeXnote}
  There is also \cs{ZeckBList} which produces the indices as a sequence of
  braced items.  To manipulate conveniently such outputs you need macros from
  \xinttoolsname or from \LaTeX3.  It is easier to use the powerful
  |\xintiieval| interface such as for example:
\begin{everbatim*}
The first five indices are \xintiieval{*zeck(123456789123456789123456789)[:5]}.
\end{everbatim*}

\begin{everbatim*}
The Zeckendorf representation of 123456789123456789123456789 uses
\xintiieval{len(zeck(123456789123456789123456789))} Fibonacci numbers.
\end{everbatim*}
\end{TeXnote}

\item[\cs{ZeckWord}]  This computes the Zeck representation as a binary word.
  The input is only \emph{f}-expanded, if you need it
  to be an expression you must wrap it in |\xintiieval|.

  \textbf{IMPORTANT:} The input \textbf{must be positive}.  No check
  is made that this is the case.

  \textbf{Note:} Input must not have more than a few thousand decimal digits.

\begin{everbatim*}
\ZeckWord{123456789}
\end{everbatim*}
\begin{everbatim*}
\ZeckWord{\xintiieval{2^40}}
\end{everbatim*}

  As \TeX{} does not by default split long strings of digits at the line ends,
  we gave so far only some small examples.  See \xintname or \bnumexprname
  documentations for a \cs{printnumber} macro able to add linebreaks.  Using 
  such an auxiliary (a bit refined) we can for example obtain this:
\begin{everbatim}
\ZeckWord{\xintiieval{2^100}}
\end{everbatim}

\vspace{-4pt}
\begingroup\colorlet{shortverbcolor}{everbatimxfgcolor}
\expandafter|\expanded{\ZeckWord{\xintiieval{2^100}}}|
\endgroup

Compare the above with the list of indices in the Zeckendorf representation:
\textcolor{everbatimxfgcolor}{\ZeckZeck{\xintiieval{2^100}}}.

\item[\cs{ZeckNFromIndices}] This computes an integer from a list of (comma
  separated) indices.  These indices do not have to be positive, their order
  is indifferent and they can be repeated or differ by only one unit.  The
  list is allowed to be empty. Contiguous commas (or commas separated only by
  space characters) act as a single one, a final comma is tolerated.  A new
  \emph{f}-expansion is done at each item, they can be (\emph{f}-expandable)
  macros.

\begin{everbatim*}
\ZeckNFromIndices{}\newline
\ZeckNFromIndices{100, ,,, 90, 80, 70, 60, 50, 40, 30 , , ,,,}
\end{everbatim*}
\begin{everbatim*}
\ZeckIndices{357128524055170099155}
\end{everbatim*}
\begin{everbatim*}
\ZeckIndices{\ZeckNFromIndices{100, 90, 80, 70, 60, 50, 40, 30}}
\end{everbatim*}
\begin{everbatim*}
\ZeckNFromIndices{3,-1,4,-1,5,-9,2,-6,5,-3}
\end{everbatim*}

There is no associated |\xintiieval| function (currently) but it is a one-liner
in its syntax:
\begin{everbatim*}
\xintiieval{add(fib(i), i= 100, 90, 80, 70, 60, 50, 40, 30)}
\end{everbatim*}
\begin{everbatim*}
\xintiieval{add(fib(i), i= 3, -1, 4, -1, 5, -9, 2, -6, 5, -3)}
\end{everbatim*}
\begin{everbatim*}
\xintiieval{add(fib(i), i = *zeck(1e60))}
\end{everbatim*}

\item[\cs{ZeckNFromWord}] This computes a positive integer from a binary word.
  The word can be arbitrary apart from not being empty.

\begin{everbatim*}
\ZeckNFromWord{1}, \ZeckNFromWord{11}, \ZeckNFromWord{111},
\ZeckNFromWord{1111}, \ZeckNFromWord{11111}
\end{everbatim*}
\begin{everbatim*}
\ZeckNFromWord{\xintReplicate{30}{10}}
\end{everbatim*}
\begin{everbatim*}
\ZeckWord{4052739537880}
\end{everbatim*}

\item[\cs{ZeckKMul}] This computes the Knuth multiplication of its two
  \textbf{positive} integer arguments.  The two arguments are only
  \emph{f}-expanded, you need to wrap each in an |\xintiieval| if
  it is an expression.

  The syntax of |\xintiieval| is extended via addition of a |$| %$
  infix operator, which gives a more convenient interface.

\begin{everbatim*}
\ZeckKMul{100}{200}
\end{everbatim*}
\begin{everbatim*}
\ZeckKMul{\ZeckKMul{100}{200}}{300}
\end{everbatim*}
\begin{everbatim*}
\ZeckKMul{100}{\ZeckKMul{200}{300}}
\end{everbatim*}

\begin{everbatim*}
\xintiieval{100 $ 200, (100 $ 200) $ 300, 100 $ (200 $ 300)}
\end{everbatim*}

  The implementation is done via the Knuth definition: each operand is
  converted to a Zeckendorf representation, the indices are added and the sum
  of Fibonacci numbers is computed.  Let us mention here that we could have
  defined a |knuth()| function easily using the powerful |\xintiieval| syntax:%
%
\footnote{We could not have used \expandafter|\string\xintdefiifunc|
  here to define |knuth()|, so we used the \expandafter|\string\xintNewFunction|
  interface.  The sole inconvenient is that when using |knuth()| it is
  as if we injected by hand the replacement expression, which will have
  to be parsed by \cs{xintiieval}.}
%
\begin{everbatim*}
\xintNewFunction{knuth}[2]
   {add(fib(x), x = flat(ndmap(+, *zeck(#1); *zeck(#2);)))}
\xintiieval{knuth(100,200), knuth(knuth(100,200),300),
                            knuth(100,knuth(200,300))}
\end{everbatim*}

The advantage of knowing this is that we can now check that our intuition
about what happens when we compute |(a $ b) $ c|, which Knuth proved to be the
same as |a $ (b $ c)|, is valid:
\begin{everbatim*}
\xintNewFunction{knuththree}[3]
 {add(fib(x), x= flat(ndmap(+, *zeck(#1); *zeck(#2); *zeck(#3);)))}
\xintiieval{knuththree(100, 200, 300)}
\newline
\xintiieval{knuththree(1000, 2000, 3000), (1000 $ 2000) $ 3000,
                                          1000 $ (2000 $ 3000)}
\end{everbatim*}

\item[\cs{ZeckSetAsKnuthOperator}] This takes as input one or more characters and
  make them work as infix operator inside of |\xintiieval| to compute the
  Knuth multiplication.  The pre-defined use of |$| for this will not
  be canceled.  You need to also do |\ZeckDeleteOperator{$}| if you want this
  meaning of |$| to be lost.  In general repeated usage will only extend the
  list of operators doing the Knuth multiplication without removing the
  previously defined ones, % $
  except if |\ZeckDeleteOperator| is used for them.

    Careful! If you do for example |\ZeckSetAsKnuthOperator{*}| the meaning of |*|
    in |\xintiieval| will be overwritten and is not recoverable (except for
    experts, of which there is only one).  But if done inside a group or
    environment, the former meaning will be recovered on exit.


  A possible choice is to use rather |$$|.  This may help avoiding syntax
  highlighting problems in your editor (or make them worse as I am currently
  experimenting while writing this).

\begin{everbatim*}
\ZeckSetAsKnuthOperator{$$}
\xintiieval{100 $$ 200 $$ 300}
\end{everbatim*}

  There are a few important points to stress:
  \begin{itemize}[nosep]
  \item You can use a letter such as |o| as operator but it then must be used
    prefixed by |\string| which is not convenient:
\begin{everbatim*}
\ZeckSetAsKnuthOperator{o}
\xintiieval{100 \string o 200 \string o 300}
\end{everbatim*}
  \item In case your document uses Babel and  some characters are catcode
    active, you must prefix them with |\string| in the argument of 
    |\ZeckSetAsKnuthOperator|, but |\string| is then \emph{unneeded}
    inside |\xintiieval| (requires \xintexprname |1.4n| or later).
  \item \textbf{There is no warning} if you overrule an already existing
    operator, hence break some parts of the syntax. For example |++| is
    already an operator. Worse, some such multi-character
    operators are defined for internal reasons and not documented at user
    level, so bad surprises can happen.
  \item Trying to set the period |.| as Knuth operator remains without
    effect.  Avoid |..| which already exists in the syntax, but you can use
    |...|.  However maybe it will be added to the syntax at some point! 
  \item Do not use the semicolon |;| as this will break for example
    its use as in the definition of
    |knuththree()| in the previous item.  You can (attow) choose |;;|.
\begin{everbatim*}
\ZeckSetAsKnuthOperator{;;}
\xintiieval{100 ;; 200 ;; 300}
\end{everbatim*}

    Note that |;*| is risky, see again the |knuththree()| example in previous
    item (spaces are ignored so |; *| is like |;*|.)
  \end{itemize}
   All such choices however may break if \xintexprname extends someday its
   list of built-in operators.  By the way this package is in \emph{alpha}
   stage and it is possible that |$| will be replaced by something else in
   future, and perhaps be used upstream in the syntax for some other task.

 \item[\cs{ZeckIndexedSum}] This is a utility which produces (expandably) |F_a
   + F_{a'} +| \dots{} where |a, a', ...| are the Zeckendorf indices in
   decrasing order and the Fibonacci numbers are represented by the letter |F|
   and the index as subscript.  Can only be used from inside math mode.
\begin{everbatim*}
$\ZeckIndexedSum{100000000000000}$.
\end{everbatim*}

   The |+| is actually injected by |\ZeckIndexedSumSep| which defaults
   to mean |{+\allowbreak}|, so that as shown above a linebreak can be
   inserted by \TeX.

 \item[\cs{ZeckExplicitSum}] This is a utility which produces (expandably)
   |F_a + F_{a'} +| \dots{} where |a, a', ...| are the Zeckendorf indices in
   decrasing order, and the Fibonacci numbers are written explicitly using
   decimal digits.  May be used outside of math mode, but there will then be
   no spacing around the |+| signs.
\begin{everbatim*}
$\ZeckExplicitSum{100000000000000}$.
\end{everbatim*}

   The |+| is actually injected by |\ZeckExplicitSumSep| which defaults
   to mean |{+\allowbreak}|, so that as shown above a linebreak can be
   inserted by \TeX.

   However, as one can see above and was already mentioned, \TeX{} and
   \LaTeX{} do not know out-of-the-box to split strings of digits at line
   endings.  Hence the first line is squeezed, which is not pleasing, and a
   number extends nevertheless into the margin.  The actual printing (and
   computation from the index) of the Fibonacci number is done via |\ZeckExplicitOne|
   whose default definition is to be an alias of |\ZeckTheFN|.

   So if we
   redefine for example this way
   \centeredline{|\renewcommand\ZeckExplicitOne[1]{F_{\ZeckTheFN{#1}}}|}
   we will simply reconstruct what |\ZeckIndexedSum| does.
   Or, with the help of a \xinttoolsname utility we can inject breakpoints
   in between digits:
\begin{everbatim*}
\renewcommand\ZeckExplicitOne[1]
   {\xintListWithSep{\allowbreak}{\ZeckTheFN{#1}}}
$\ZeckExplicitSum{100000000000000}$.
\end{everbatim*}

\begingroup
\colorlet{shortverbcolor}{everbatimxfgcolor}
\mathsurround0pt
\def\ZeckExplicitSumSep{{}+{}}
\def\ZeckExplicitOne#1{$\expandafter|\expanded{\ZeckTheFN{#1}}|$}%
% exit math mode because we use \ttfamily
   Expert \LaTeX{} users will know how to achieve a result
   such as this one, which pleasantly decorate the linebreaks:

$\ZeckExplicitSum{100000000000000}$.\par
\endgroup
   
\end{description}

\section{Use with Plain \eTeX{}}
\label{sec:plain}

You will need to input the core code using:
\centeredline{|\input zeckendorfcore.tex|}
\textbf{IMPORTANT:} after this |\input|, the catcode regimen is a specific
one (for example |_|, |:|, and |^| all have catcode letter).  So, you will
probably want to emit |\ZECKrestorecatcodes| immediately after this
import, it will reset all modified catcodes to their values as prior to the
import.

Then you can use the exact same interface as described in the previous
section.

\begingroup
\small
% Systématiquement à devoir hacker pour avoir quelque chose
% qui fonctionne correctement
\makeatletter\def\size@section{\normalsize} \makeatother

\section{Changes}
\label{sec:changes}

\begin{description}
\input zeckendorfchanges.tex
\end{description}

\section{License}
\label{sec:license}

\makeatletter
% 2025/10/04
% attention au © qui pose problème avec ts1newtxttz.fd bogué
% j'ai fait le report à MS mais peut-être mon mail il y a plusieurs
% mois a fini dans le spam.  Donc ici ne pas oublier le \ttfamily.
%
% 2025/10/06 le © est cassé avec xelatex + newtxtt. Sur un mwe
% si on utilise newtxtext c'est bon, mais pas ici et je n'ai pas
% le temps de m'acharner.
\def\MacroFont{\ttfamily \baselineskip11pt }%
\begin{verbatim}
Copyright (c) 2025 Jean-François Burnol

| This Work may be distributed and/or modified under the
| conditions of the LaTeX Project Public License 1.3c.
| This version of this license is in

>   <http://www.latex-project.org/lppl/lppl-1-3c.txt>

| and version 1.3 or later is part of all distributions of
| LaTeX version 2005/12/01 or later.

This Work has the LPPL maintenance status "author-maintained".

The Author and Maintainer of this Work is Jean-François Burnol.

This Work consists of the main source file and its derived files

    zeckendorf.dtx,
    zeckendorfcore.tex, zeckendorf.tex, zeckendorf.sty,
    README.md, zeckendorf-doc.tex, zeckendorf-doc.pdf

\end{verbatim}
\endgroup

\StopEventually{\end{document}\endinput}

% provisoire
\def\csh{\cs}

% workaround
% https://github.com/latex3/latex2e/issues/563
\makeatletter
\AddToHook{env/macrocode/begin}{\partopsep0pt\relax}
\AddToHook{env/macrocode/after}{\@nobreakfalse}
\makeatother
\newgeometry{hscale=0.75,vscale=0.75}% ATTENTION \newgeometry fait
                                % un reset de vscale si on ne le
                                % précise pas ici !!!

\MakePercentIgnore
\AddToHook{env/macrocode/begin}{\bfseries}
%
% \catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
% \let</dtx>\relax
% \def<*core>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</dtx>
%<*core>
% \etocdepthtag.toc{code}
% \part{Commented source code}
% \localtableofcontents
%
% \label{part:zeckendorfcode}
% \etocmarkbothnouc{\zecknameuserdoc \hyperref[sec:zeckendorfcode]{implementation}}
% \section{Core code}
% \label{sec:code:core}
% \etocignoredepthtags
% \etocsetnexttocdepth{subsection}
% \localtableofcontents
%
% A general remark is that expandable macros (usually) \emph{f}-expand their
% arguments, and most are \emph{f}-expandable.  This \emph{f}-expandability is
% achieved via \cs{expanded} triggers, diverging a bit from the overall style
% of the \xintname codebase (which predates availability of \cs{expanded}).
%
% Extracts to |zeckendorfcore.tex|.
%
% \subsection{Loading \xintexprname and setting catcodes}
%    \begin{macrocode}
\input xintexpr.sty
\wlog{Package: zeckendorfcore 2025/10/06 v0.9alpha (JFB)}%
\edef\ZECKrestorecatcodes{\XINTrestorecatcodes}%
\def\ZECKrestorecatcodesendinput{\ZECKrestorecatcodes\noexpand\endinput}%
\XINTsetcatcodes%
%    \end{macrocode}
% Small helpers related to \cs{expanded}-based methods.  But the package only
% has a few macros and these helpers are used only once or twice, some macros
% needing their own terminators due to various optimizations in the code
% organization.
%    \begin{macrocode}
\def\zeck_abort#1\xint:{{}}%
\def\zeck_done#1\xint:{\iffalse{\fi}}%
%    \end{macrocode}
% \subsection{Support for computing Fibonacci numbers:
% \csh{ZeckTheFN}, \csh{ZeckTheFSeq}}
% The multiplicative algorithm is as in the \bnumexprname manual (at
% |1.7b|) termination is different.
%
% |\Zeck@FPair| and |\Zeck@@FPair| are not public interface.
% The former is a wrapper of the latter to handle negative or zero
% argument.
%
% The public |\ZeckTheFN| uses the |\Zeck@FPair| which accepts a negative
% or zero argument.  The non public |\Zeck@@FN| uses |\Zeck@@FPair|
% and is thus limited to positive argument, also it remains in |\xintexpr|
% encapsulated format requiring |\xintthe| for explicit digits.
%    \begin{macrocode}
\def\Zeck@FPair#1{\expandafter\zeck@fpair\the\numexpr #1.}%
\def\zeck@fpair #1{%
   \xint_UDzerominusfork
      #1-\zeck@fpair_n
      0#1\zeck@fpair_n
      0-\zeck@fpair_p
   \krof #1%
}%
\def\zeck@fpair_p #1.{\Zeck@@FPair{#1}}%
\def\zeck@fpair_n #1.{%
    \ifodd#1 \expandafter\zeck@fpair_ei\else\expandafter\zeck@fpair_eii\fi
    \romannumeral`&&@\Zeck@@FPair{1-#1}%
}%
\def\zeck@fpair_ei{\expandafter\zeck@fpair_fi}%
\def\zeck@fpair_eii{\expandafter\zeck@fpair_fii}%
\def\zeck@fpair_fi#1;#2;{%
    \romannumeral0\xintiiexpro #2\expandafter\relax\expandafter;%
    \romannumeral0\xintiiexpro -#1\relax;%
}%
\def\zeck@fpair_fii#1;#2;{%
    \romannumeral0\xintiiexpro -#2\expandafter\relax\expandafter;%
    #1;%
}%
\def\Zeck@@FPair#1{%
    \expandafter\Zeck@start
    \romannumeral0\xintdectobin{\the\numexpr#1\relax};%
}%
\def\Zeck@start 1#1{%
    \csname Zeck@#1\expandafter\endcsname
    \romannumeral0\xintiiexpro 1\expandafter\relax\expandafter;%
    \romannumeral0\xintiiexpro 0\relax;%
}%
\expandafter\def\csname Zeck@0\endcsname #1;#2;#3{%
    \csname Zeck@#3\expandafter\endcsname
    \romannumeral0\xintiiexpro (#1+2*#2)*#1\expandafter\relax\expandafter;%
    \romannumeral0\xintiiexpro #1*#1+#2*#2\relax;%
}%
\expandafter\def\csname Zeck@1\endcsname #1;#2;#3{%
    \csname Zeck@#3\expandafter\endcsname
    \romannumeral0\xintiiexpro 2*(#1+#2)*#1+#2*#2\expandafter\relax\expandafter;%
    \romannumeral0\xintiiexpro (#1+2*#2)*#1\relax;%
}%
\expandafter\let\csname Zeck@;\endcsname\empty
%    \end{macrocode}
% For individual Fibonacci numbers, we have non public |\Zeck@@FN| and public
% |\ZeckTheFN|.
%    \begin{macrocode}
\def\Zeck@@FN{\expandafter\zeck@@fn\romannumeral`&&@\Zeck@@FPair}%
\def\zeck@@fn#1;#2;{#1}%
\def\ZeckTheFN{\xintthe\expandafter\zeck@@fn\romannumeral`&&@\Zeck@FPair}%
%    \end{macrocode}
% The computation of stretches of Fibonacci numbers is not needed for the
% package, but is provided for user convenience.  This is lifted from the
% development version of the |\xintname| user manual, which refactored a bit
% the code which has been there for over ten years.  As we want to add a
% |fibseq()| function to |\xintiieval|, it is better to make it
% \emph{f}-expandable.
%
% Here we also handle negative arguments but still require the
% second argument to be larger (more positive) than the first.
%    \begin{macrocode}
\def\ZeckTheFSeq#1#2{%#1=starting index, #2>#1=ending index
    \expanded\bgroup\expandafter\ZeckTheF@Seq
    \the\numexpr #1\expandafter.\the\numexpr #2.%
}%
%    \end{macrocode}
% The |#1+1| is because |\Zeck@FPair{N}| expands to |F_{N};F_{N-1};|,
% so here we will have |F_{A+1},F_{A};| as starting point.
% We want up to |F_B|.
% If |B=A+1| there will be nothing to do.
%    \begin{macrocode}
\def\ZeckTheF@Seq #1.#2.{%
    \expandafter\ZeckTheF@Seq@loop
    \the\numexpr #2-#1-1\expandafter.\romannumeral0\Zeck@FPair{#1+1}%
}%
%    \end{macrocode}
% Now leave in stream one coefficient, test if we have reached B
% and until then apply standard Fibonacci recursion.
% We insert |\xintthe| although not needed for typesetting but
% this is useful for matters of defining an associated |fibseq()|
% function. 
%    \begin{macrocode}
\def\ZeckTheF@Seq@loop #1.#2;#3;{% standard Fibonacci recursion
    {\xintthe#3}\ifnum #1=\z@ \expandafter\ZeckTheF@Seq@end\fi
    \expandafter\ZeckTheF@Seq@loop
    \the\numexpr #1-1\expandafter.%
    \romannumeral0\xintiiexpro #2+#3\relax;#2;%
}%
\def\ZeckTheF@Seq@end#1;#2;{{\xintthe#2}\iffalse{\fi}}%
%    \end{macrocode}
% \subsection{\csh{ZeckNearIndex}, \csh{ZeckIndex}}
% If the ratio of logarithms was the exact mathematical value it would be
% certain (via rough estimates valid at least for say $x\geq20$, and manual
% checks for lower $x$'s) that its integer rounding gives an integer |K| such
% that either |K| or |K-1| is the largest index |J| with $F_J\leq x$.  But the
% computation is done with only about 8 or 9 digits of precision.  So
% certainly this assumption would fail for $x$ having more than one billion
% decimal digits, or even perhaps would become a bit risky with an input
% having ten million digits.  But this is no worry for us because we can not
% hope to compute with more than say about 20000 digits.  We are not at all
% worried about the rounding to an integer not giving the exact rounding of
% the theoretical exact ratio of logarithms, or perhaps not being the correct
% rounding of the actually computed numerical approximation.  Indeed prior to
% rounding the computed numerical approximation, we are close to the exact
% theoretical one.  If the numerical rounding differs from the exact rounding
% of the exact value, it must be that $x$ is about mid-way (in log scale)
% between two consecutive Fibonacci numbers.  So the numerical rounding can
% not be anything else than either |J| or |J+1|.
%
% (the argument is more subtle than it looks, it is important as it
% means we do not have to add extraneous checks).
%
% The formula with macros was obtained via an |\xintdeffloatfunc| and
% |\xintverbosetrue| after having set |\xintDigits*| to \dgts{8}, and then we
% optimized a bit manually.  The advantage here is that we don't have to set
% ourself |\xintDigits| and later restore it.  We can not use |\XINTinFloatDiv| or
% |\XINTinFloatMul| if we don't set |\xintDigits| (which is user customizable)
% because they hardcode usage of |\XINTdigits|.
%    \begin{macrocode}
\def\ZeckNearIndex#1{\xintiRound{0}{%
    \xintFloatDiv[8]{\PoorManLogBaseTen{\xintFloatMul[8]{2236068[-6]}{#1}}}%
                    {20898764[-8]}%
                          }%
}%
%    \end{macrocode}
% Now we compute the actual maximal index.  This macro is only for user
% interface, because when obtaining the Zeckendorf representation via the
% greedy algorithm, we will want for efficienty to not discard the computed
% pair of Fibonacci numbers, but proceed using it.
%    \begin{macrocode}
\def\ZeckIndex{\expanded\zeckindex}%
\def\zeckindex#1{\expandafter\zeckindex_fork\romannumeral`&&@#1\xint:}%
\def\zeckindex_fork#1{%
  \xint_UDzerominusfork
    #1-\zeck_abort
    0#1\zeck_abort
    0-{\zeckindex_a#1}%
  \krof
}%
\def\zeckindex_a #1\xint:{%
    \expandafter\zeckindex_b
    \the\numexpr\ZeckNearIndex{#1}\xint:#1\xint:
}%
\def\zeckindex_b #1\xint:{%
    \expandafter\zeckindex_c
    \romannumeral`&&@\Zeck@@FPair{#1}#1\xint:
}%
\def\zeckindex_c #1;#2;#3\xint:#4\xint:{%
    \xintiiifGt{\xintthe#1}{#4}%
       {{\expandafter}\the\numexpr#3-1\relax}%
       {{}#3}%
}%
%    \end{macrocode}
% \subsection{\csh{ZeckIndices}, \csh{ZeckZeck}}
% As explained at start of code comments, I decided when packaging the whole
% thing to make macros \emph{f}-expandable via \cs{expanded}-trigger, not
% \cs{romannumeral}.
%
% This and other macros start by computing the max index.  It then subtracts
% the Fibonacci number from the input and loops.
%    \begin{macrocode}
\def\ZeckIndices{\expanded\zeckindices}%
\let\ZeckZeck\ZeckIndices
\def\zeckindices#1{\expandafter\zeckindices_fork\romannumeral`&&@#1\xint:}%
\def\zeckindices_fork#1{%
  \xint_UDzerominusfork
    #1-\zeck_abort
    0#1\zeck_abort
    0-{\bgroup\zeckindices_a#1}%
  \krof
}%
\def\zeckindices_a #1\xint:{%
    \expandafter\zeckindices_b
    \the\numexpr\ZeckNearIndex{#1}\xint:#1\xint:
}%
\def\zeckindices_b #1\xint:{%
    \expandafter\zeckindices_c
    \romannumeral`&&@\Zeck@@FPair{#1}#1\xint:
}%
\def\zeckindices_c #1;#2;#3\xint:#4\xint:{%
    \xintiiifGt{\xintthe#1}{#4}\zeckindices_A\zeckindices_B
    #1;#2;#3\xint:#4\xint:
}%
%    \end{macrocode}
% There is a slight annoyance here which is that we have to use the
% |\xintthe...| macros to have explicit digits so that we can test (and using
% higher level interface unavoidably boils down to looking at explicit digit
% tokens) if the first one is zero.  But alas, the \xintexprname manual has
% documented things such as |\xintiiexprPrintOne| as being customizable, so
% there is a potentiality here for user modifications causing a crash, if a
% custom |\xintiiexprPrintOne| prints |Z| or some other symbol in case of the
% zero value...  We do have at our disposal |\xintthebareiieval| but it needs
% one more brace stripping step.  So some |\xinttheunbracedbareiieval| is needed
% upstream and when this is done the code here will get updated.
%    \begin{macrocode}
\def\zeckindices_A#1;#2;#3\xint:#4\xint:{%
    \the\numexpr#3-1\relax
    \expandafter\zeckindices_loop
    \romannumeral`&&@\xinttheiiexpr #4-#2\relax\xint:
}%
\def\zeckindices_B#1;#2;#3\xint:#4\xint:{%
    #3%
    \expandafter\zeckindices_loop
    \romannumeral`&&@\xinttheiiexpr #4-#1\relax\xint:
}%
\def\zeckindices_loop#1{%
    \xint_UDzerofork#1\zeck_done 0{, \zeckindices_a#1}\krof
}%
%    \end{macrocode}
% \subsection{\csh{ZeckBList}}
% This is the variant which produces the results as a sequence of braced
% indices.  Useful as support for a |zeck()| function.
%
% Originally in \xintname, \xinttoolsname, the term ``list'' is used for
% braced items.  In the user manual of this package I have been using ``list''
% more colloquially for comma separated values.  Here I stick with \xintname
% conventions but use |BList| (short for ``list of Braced items'') and not only
% ``List'' in the name.
%    \begin{macrocode}
\def\ZeckBList{\expanded\zeckblist}%
\def\zeckblist#1{\expandafter\zeckblist_fork\romannumeral`&&@#1\xint:}%
\def\zeckblist_fork#1{%
  \xint_UDzerominusfork
    #1-\zeck_abort
    0#1\zeck_abort
    0-{\bgroup\zeckblist_a#1}%
  \krof
}%
\def\zeckblist_a #1\xint:{%
    \expandafter\zeckblist_b
    \the\numexpr\ZeckNearIndex{#1}\xint:#1\xint:
}%
\def\zeckblist_b #1\xint:{%
    \expandafter\zeckblist_c
    \romannumeral`&&@\Zeck@@FPair{#1}#1\xint:
}%
\def\zeckblist_c #1;#2;#3\xint:#4\xint:{%
    \xintiiifGt{\xintthe#1}{#4}\zeckblist_A\zeckblist_B
    #1;#2;#3\xint:#4\xint:
}%
\def\zeckblist_A#1;#2;#3\xint:#4\xint:{%
    {\the\numexpr#3-1\relax}%
    \expandafter\zeckblist_loop
    \romannumeral`&&@\xinttheiiexpr #4-#2\relax\xint:
}%
\def\zeckblist_B#1;#2;#3\xint:#4\xint:{%
    {#3}%
    \expandafter\zeckblist_loop
    \romannumeral`&&@\xinttheiiexpr #4-#1\relax\xint:
}%
\def\zeckblist_loop#1{\xint_UDzerofork#1\zeck_done 0{\zeckblist_a#1}\krof}%
%    \end{macrocode}
% \subsection{\csh{ZeckIndexedSum}, \csh{ZeckExplicitSum}}
% They are expandable, but need \emph{x}-expansion.  The first one assumes it
% expands in math mode.  We use |\sb| because the current catcode of |_| is letter,
% and using |\sb| spares us some juggling.
%    \begin{macrocode}
\def\ZeckIndexedSumSep{+\allowbreak}%
\let\ZeckExplicitSumSep\ZeckIndexedSumSep
\def\ZeckExplicitOne{\xintthe\Zeck@@FN}%
\def\ZeckIndexedSum#1{%
    \expandafter\zeckindexedsum\expanded\zeckindices{#1},;%
}%
\def\zeckindexedsum#1{%
    \if,#1\expandafter\xint_gob_til_sc\fi \zeckindexedsum_a#1%
}%
\def\zeckindexedsum_a#1,{F\sb{#1}\zeckindexedsum_b}%
\def\zeckindexedsum_b #1{%
    \if;#1\expandafter\xint_gob_til_sc\fi
    \ZeckIndexedSumSep\zeckindexedsum_a#1%
}%
\def\ZeckExplicitSum#1{%
    \expandafter\zeckexplicitsum\expanded\zeckindices{#1},;%
}%
\def\zeckexplicitsum#1{%
    \if,#1\expandafter\xint_gob_til_sc\fi \zeckexplicitsum_a#1%
}%
\def\zeckexplicitsum_a#1,{\ZeckExplicitOne{#1}\zeckexplicitsum_b}%
\def\zeckexplicitsum_b #1{%
    \if;#1\expandafter\xint_gob_til_sc\fi
    \ZeckExplicitSumSep\zeckexplicitsum_a#1%
}%
%    \end{macrocode}
% \subsection{\csh{ZeckWord}}
% This is slightly more complicated than \cs{ZeckIndices} and \cs{ZeckBList}
% because we have to keep track of the previous index to know how many zeros
% to inject.
%    \begin{macrocode}
\def\ZeckWord{\expanded\zeckword}%
\def\zeckword#1{\expandafter\zeckword_fork\romannumeral`&&@#1\xint:}%
\def\zeckword_fork#1{%
  \xint_UDzerominusfork
    #1-\zeck_abort
    0#1\zeck_abort
    0-{\bgroup\zeckword_a#1}%
  \krof
}%
\def\zeckword_a #1\xint:{%
    \expandafter\zeckword_b\the\numexpr\ZeckNearIndex{#1}\xint:
    #1\xint:
}%
\def\zeckword_b #1\xint:{%
    \expandafter\zeckword_c\romannumeral`&&@\Zeck@@FPair{#1}#1\xint:
}%
\def\zeckword_c #1;#2;#3\xint:#4\xint:{%
    \xintiiifGt{\xintthe#1}{#4}\zeckword_A\zeckword_B
    #1;#2;#3\xint:#4\xint:
}%
\def\zeckword_A#1;#2;#3\xint:#4\xint:{%
    \expandafter\zeckword_d
    \romannumeral`&&@\xinttheiiexpr#4-#2\expandafter\relax\expandafter\xint:
    \the\numexpr#3-1.%
}%
\def\zeckword_B#1;#2;#3\xint:#4\xint:{%
    \expandafter\zeckword_d
    \romannumeral`&&@\xinttheiiexpr#4-#1\relax\xint:
    #3.%
}%
\def\zeckword_d #1%
    {\xint_UDzerofork#1\zeckword_done0{1\zeckword_e}\krof #1}%
\def\zeckword_done#1\xint:#2.{1\xintReplicate{#2-2}{0}\iffalse{\fi}}%
\def\zeckword_e #1\xint:{%
    \expandafter\zeckword_f\the\numexpr\ZeckNearIndex{#1}\xint:
    #1\xint:
}%
\def\zeckword_f #1\xint:{%
    \expandafter\zeckword_g\romannumeral`&&@\Zeck@@FPair{#1}#1\xint:
}%
\def\zeckword_g #1;#2;#3\xint:#4\xint:{%
    \xintiiifGt{\xintthe#1}{#4}\zeckword_gA\zeckword_gB
    #1;#2;#3\xint:#4\xint:
}%
\def\zeckword_gA#1;#2;#3\xint:#4\xint:{%
    \expandafter\zeckword_h
    \the\numexpr#3-1\expandafter.%
    \romannumeral`&&@\xinttheiiexpr #4-#2\relax\xint:
}%
\def\zeckword_gB#1;#2;#3\xint:#4\xint:{%
    \expandafter\zeckword_h
    \the\numexpr#3\expandafter.%
    \romannumeral`&&@\xinttheiiexpr #4-#1\relax\xint:
}%
\def\zeckword_h #1.#2\xint:#3.{%
    \xintReplicate{#3-#1-1}{0}%
    \zeckword_d #2\xint:#1.%
}%
%    \end{macrocode}
% \subsection{The Knuth Multiplication: \csh{ZeckKMul}}
% Here a |\romannumeral0| trigger is used to match |\xintiisum|.  Although
% it induces defining one more macro we obide by the general coding style of
% \xintname with a CamelCase then a lowercase macro, rather than having them 
% merged as only one.
%    \begin{macrocode}
\def\ZeckKMul{\romannumeral0\zeckkmul}%
\def\zeckkmul#1#2{\expandafter\zeckkmul_a
                  \expanded{\ZeckIndices{#1}%
                            ,;%
                            \ZeckIndices{#2}%
                            ,,}%
}%
%    \end{macrocode}
% The space token at start of |#2| after first one is not a problem
% as it ends up in a |\numexpr| anyhow.
%    \begin{macrocode}
\def\zeckkmul_a{\expandafter\xintiisum\expanded{{\iffalse}}\fi
                \zeckkmul_b}%
\def\zeckkmul_b#1;#2,{%
    \if\relax#2\relax\expandafter\zeckkmul_done\fi
    \zeckkmul_c{#2}#1,\zeckkmul_b#1;%
}%
\def\zeckkmul_c#1#2,{%
    \if\relax#2\relax\expandafter\xint_gobble_iii\fi
    {\xintthe\Zeck@@FN{#1+#2}}\zeckkmul_c{#1}%
}%
\def\zeckkmul_done#1;{\iffalse{{\fi}}}%
%    \end{macrocode}
% \subsection{\csh{ZeckNFromIndices}}
% Spaces before commas are not a problem they disappear in |\numexpr|.
%
% Each item is \emph{f}-expanded to check not empty, but perhaps we could skip
% expanding, as they end up in |\numexpr|.  Advantage of expansion of each
% item is that at any location we can generate multiple indices if desired.
%    \begin{macrocode}
\def\ZeckNFromIndices{\romannumeral0\zecknfromindices}%
\def\zecknfromindices#1{\expandafter\zecknfromindices_a\romannumeral`&&@#1,;}%
\def\zecknfromindices_a{\expandafter\xintiisum\expanded{{\iffalse}}\fi
                  \zecknfromindices_b
}%
\def\zecknfromindices_b#1{%
    \if;#1\xint_dothis\zecknfromindices_done\fi
    \if,#1\xint_dothis\zecknfromindices_skip\fi
    \xint_orthat\zecknfromindices_c #1%
}%
\def\zecknfromindices_c #1,{%
    {\ZeckTheFN{#1}}\expandafter\zecknfromindices_b\romannumeral`&&@%
}%
\def\zecknfromindices_skip,{\expandafter\zecknfromindices_b\romannumeral`&&@}%
\def\zecknfromindices_done;{\iffalse{{\fi}}}%
%    \end{macrocode}
% \subsection{\csh{ZeckNFromWord}}
% The |\xintreversedigits| will \emph{f}-expand its argument.
%    \begin{macrocode}
\def\ZeckNFromWord{\romannumeral0\zecknfromword}%
\def\zecknfromword#1{%
    \expandafter\zecknfromword_a\romannumeral0\xintreversedigits{#1};%
}%
\def\zecknfromword_a{%
    \expandafter\xintiisum\expanded{{\iffalse}}\fi\zecknfromword_b 2.%
}%
\def\zecknfromword_b#1.#2{%
    \if;#2\expandafter\zecknfromword_done\fi
    \if#21{\xintthe\Zeck@@FN{#1}}\fi
    \expandafter\zecknfromword_b\the\numexpr#1+1.%
}%
\def\zecknfromword_done#1.{\iffalse{{\fi}}}%
%    \end{macrocode}
% \subsection{Extension of the \cs{xintiieval} syntax with fib(), fibseq(),
% zeck() and zeckindex() functions}
% |fib()| and |fibseq()|  accept negative arguments, but |fibseq(a,b)| must be
% with |b>a|, else falls into an infinite loop.  |zeck()| and |zeckindex()|
% require, but do not check, that their argument is positive.
%
% We also add support for these functions to |\xinteval| and |\xintfloateval|.
% Arguments are then truncated (not rounded) to integers.
%    \begin{macrocode}
\def\XINT_iiexpr_func_fib #1#2#3%
{%
    \expandafter #1\expandafter #2\expandafter{%
    \romannumeral`&&@\XINT:NEhook:f:one:from:one
    {\romannumeral`&&@\ZeckTheFN#3}}%
}%
\def\ZeckTheFNnum#1{\ZeckTheFN{\xintNum{#1}}}%
\def\XINT_expr_func_fib #1#2#3%
{%
    \expandafter #1\expandafter #2\expandafter{%
    \romannumeral`&&@\XINT:NEhook:f:one:from:one
    {\romannumeral`&&@\ZeckTheFNnum#3}}%
}%
\let\XINT_flexpr_func_fib\XINT_expr_func_fib
\def\XINT_iiexpr_func_fibseq #1#2#3%
{%
    \expandafter #1\expandafter #2\expandafter{%
    \romannumeral`&&@\XINT:NEhook:f:one:from:two
    {\romannumeral`&&@\ZeckTheFSeq#3}}%
}%
\def\ZeckTheFSeqnum#1#2{\ZeckTheFSeq{\xintNum{#1}}{\xintNum{#2}}}%
\def\XINT_expr_func_fibseq #1#2#3%
{%
    \expandafter #1\expandafter #2\expandafter{%
    \romannumeral`&&@\XINT:NEhook:f:one:from:two
    {\romannumeral`&&@\ZeckTheFSeqnum#3}}%
}%
\let\XINT_flexpr_func_fibseq\XINT_expr_func_fibseq
\def\XINT_iiexpr_func_zeckindex #1#2#3%
{%
    \expandafter #1\expandafter #2\expandafter{%
    \romannumeral`&&@\XINT:NEhook:f:one:from:one
    {\romannumeral`&&@\ZeckIndex#3}}%
}%
\def\ZeckIndexnum#1{\ZeckIndex{\xintNum{#1}}}%
\def\XINT_expr_func_zeckindex #1#2#3%
{%
    \expandafter #1\expandafter #2\expandafter{%
    \romannumeral`&&@\XINT:NEhook:f:one:from:one
    {\romannumeral`&&@\ZeckIndexnum#3}}%
}%
\let\XINT_flexpr_func_zeckindex\XINT_expr_func_zeckindex
\def\XINT_iiexpr_func_zeck #1#2#3%
{%
    \expandafter #1\expandafter #2\expandafter{%
    \romannumeral`&&@\XINT:NEhook:f:one:from:one
    {\romannumeral`&&@\ZeckBList#3}}%
}%
\def\ZeckBListnum#1{\ZeckBList{\xintNum{#1}}}%
\def\XINT_expr_func_zeck #1#2#3%
{%
    \expandafter #1\expandafter #2\expandafter{%
    \romannumeral`&&@\XINT:NEhook:f:one:from:one
    {\romannumeral`&&@\ZeckBListnum#3}}%
}%
\let\XINT_flexpr_func_zeck\XINT_expr_func_zeck
%    \end{macrocode}
% \subsection{Extension of the \cs{xintiieval} syntax with \textdollar{} as
% infix operator for the Knuth multiplication}
%
% Unfortunately, contrarily to \bnumexprname, \xintexprname (at
% |1.4o|) has no public interface to define an infix operator, and the macros
% it defines to that end have acquired another meaning at end of loading
% |xintexpr.sty|, so we have to copy quite a few lines of code. This is
% provisory and will be removed when xintexpr.sty will have been udpated. We
% also copy/adapt |\bnumdefinfix|.
%
% We test for existence of |\xintdefinfix| so as to be able to update upstream
% and not have to sync it immediately.  But perhaps upstream will choose some
% other name than |\xintdefinfix|...
%    \begin{macrocode}
\ifdefined\xintdefinfix
  \def\zeckdefinfix{\xintdefinfix {iiexpr}}%
\else
\ifdefined\xint_noxpd\else\let\xint_noxpd\unexpanded\fi % support old xint
\def\ZECK_defbin_c #1#2#3#4#5#6#7#8%
{%
  \XINT_global\def #1##1% \XINT_#8_op_<op>
  {%
    \expanded{\xint_noxpd{#2{##1}}\expandafter}%
    \romannumeral`&&@\expandafter#3\romannumeral`&&@\XINT_expr_getnext
  }%
  \XINT_global\def #2##1##2##3##4% \XINT_#8_exec_<op>
  {%
    \expandafter##2\expandafter##3\expandafter
      {\romannumeral`&&@\XINT:NEhook:f:one:from:two{\romannumeral`&&@#7##1##4}}%
  }%
  \XINT_global\def #3##1% \XINT_#8_check-_<op>
  {%
    \xint_UDsignfork
      ##1{\expandafter#4\romannumeral`&&@#5}%
        -{#4##1}%
    \krof
  }%
  \XINT_global\def #4##1##2% \XINT_#8_checkp_<op>
  {%
    \ifnum ##1>#6%
      \expandafter#4%
      \romannumeral`&&@\csname XINT_#8_op_##2\expandafter\endcsname
    \else
      \expandafter ##1\expandafter ##2%
    \fi
  }%
}%
%    \end{macrocode}
% ATTENTION there is lacking at end here compared to the \bnumexprname version
%           an adjustment for updating minus operator,
%           if some other right precedence than 12, 14, 17 is used.
% Doing this would requiring copying still more, so is not done.
%    \begin{macrocode}
\def\ZECK_defbin_b #1#2#3#4#5%
{%
  \expandafter\ZECK_defbin_c
  \csname XINT_#1_op_#2\expandafter\endcsname
  \csname XINT_#1_exec_#2\expandafter\endcsname
  \csname XINT_#1_check-_#2\expandafter\endcsname
  \csname XINT_#1_checkp_#2\expandafter\endcsname
  \csname XINT_#1_op_-\romannumeral\ifnum#4>12 #4\else12\fi\expandafter\endcsname
  \csname xint_c_\romannumeral#4\endcsname
  #5%
  {#1}% #8 for \ZECK_defbin_c
  \XINT_global
  \expandafter
  \let\csname XINT_expr_precedence_#2\expandafter\endcsname
      \csname xint_c_\romannumeral#3\endcsname
}%
%    \end{macrocode}
% These next two currently lifted from \bnumexprname with some adaptations,
% see previous comment about precedences.
%    \begin{macrocode}
\def\zeckdefinfix #1#2#3#4%
{%
    \edef\ZECK_tmpa{#1}%
    \edef\ZECK_tmpa{\xint_zapspaces_o\ZECK_tmpa}%
    \edef\ZECK_tmpL{\the\numexpr#3\relax}%
    \edef\ZECK_tmpL
         {\ifnum\ZECK_tmpL<4 4\else\ifnum\ZECK_tmpL<23 \ZECK_tmpL\else 22\fi\fi}%
    \edef\ZECK_tmpR{\the\numexpr#4\relax}%
    \edef\ZECK_tmpR
         {\ifnum\ZECK_tmpR<4 4\else\ifnum\ZECK_tmpR<23 \ZECK_tmpR\else 22\fi\fi}%
    \ZECK_defbin_b {iiexpr}\ZECK_tmpa\ZECK_tmpL\ZECK_tmpR #2%
    \expandafter\ZECK_dotheitselves\ZECK_tmpa\relax
  \unless\ifcsname  
    XINT_iiexpr_exec_-\romannumeral\ifnum\ZECK_tmpR>12 \ZECK_tmpR\else 12\fi
  \endcsname
    \xintMessage{zeckendorf}{Error}{Right precedence not among accepted values.}%
    \errhelp{Accepted values include 12, 14, and 17.}%
    \errmessage{Sorry, you can not use \ZECK_tmpR\space as right precedence.}%
  \fi
  \ifxintverbose
    \xintMessage{zeckendorf}{info}{infix operator \ZECK_tmpa\space
    \ifxintglobaldefs globally \fi
        does
        \xint_noxpd{#2}\MessageBreak with precedences \ZECK_tmpL, \ZECK_tmpR;}%
  \fi
}%
\def\ZECK_dotheitselves#1#2%
{%
    \if#2\relax\expandafter\xint_gobble_ii
    \else
  \XINT_global
      \expandafter\edef\csname XINT_expr_itself_#1#2\endcsname{#1#2}%
      \unless\ifcsname XINT_expr_precedence_#1\endcsname
  \XINT_global
        \expandafter\edef\csname XINT_expr_precedence_#1\endcsname
                        {\csname XINT_expr_precedence_\ZECK_tmpa\endcsname}%
  \XINT_global
        \expandafter\odef\csname XINT_iiexpr_op_#1\endcsname
                        {\csname XINT_iiexpr_op_\ZECK_tmpa\endcsname}%
      \fi
    \fi
    \ZECK_dotheitselves{#1#2}%
}%
\fi
%    \end{macrocode}
% There is no ``undefine operator'' in \bnumexprname currently.  Experimental,
% I don't want to spend too much time. I sense there is a potential problem
% with multi-character operators related to ``undoing the itselves'', because
% of the mechanism which allows to use for example |;;| as short-cut for |;;;|
% if |;;| was not pre-defined when |;;;| got defined.  To undefine |;;|, I
% would need to check if it really has been aliased to |;;;|, and I don't do
% the effort here.
%    \begin{macrocode}
\ifdefined\xintdefinfix
\else
\ifdefined\xint_noxpd\else\let\xint_noxpd\unexpanded\fi % support old xint
\def\ZECK_undefbin_b #1#2%
{%
  \XINT_global\expandafter\let
    \csname XINT_#1_op_#2\endcsname\xint_undefined
  \XINT_global\expandafter\let
    \csname XINT_#1_exec_#2\endcsname\xint_undefined
  \XINT_global\expandafter\let
    \csname XINT_#1_check-_#2\endcsname\xint_undefined
  \XINT_global\expandafter\let
    \csname XINT_#1_checkp_#2\endcsname\xint_undefined
  \XINT_global\expandafter\let
    \csname XINT_expr_precedence_#2\endcsname\xint_undefined
  \XINT_global\expandafter\let
    \csname XINT_expr_itself_#2\endcsname\xint_undefined
}%
\def\zeckundefinfix #1%
{%
    \edef\ZECK_tmpa{#1}%
    \edef\ZECK_tmpa{\xint_zapspaces_o\ZECK_tmpa}%
    \ZECK_undefbin_b {iiexpr}\ZECK_tmpa
%%  \ifxintverbose
    \xintMessage{zeckendorf}{Warning}{infix operator \ZECK_tmpa\space
        has been DELETED!}%
%%  \fi
}%
\fi
%    \end{macrocode}
% We do not define the extra |\chardef|'s as does \bnumexprname to allow more
% user choices of precedences, not only because nobody will ever use the
% feature, but also because it needs extra configuration for minus unary
% operator.  (as mentioned above)
%
% Attention, this is like |\bnumdefinfix| and thus does not have same order
% of arguments as the |\ZECK_defbin_b| above.
%    \begin{macrocode}
\zeckdefinfix{$}{\ZeckKMul}{14}{14}% $ (<-only to tame Emacs/AUCTeX highlighting)
\def\ZeckSetAsKnuthOperator#1{\zeckdefinfix{#1}{\ZeckKMul}{14}{14}}%
\def\ZeckDeleteOperator#1{\zeckundefinfix{#1}}%
%    \end{macrocode}
% ATTENTION! we leave the modified catcodes in place! (the question mark
% has regained its catcode other though).
%
% \catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
% \let</core>\relax
% \def<*interactive>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</core>
%<*interactive>
% \makeatletter\global\c@CodelineNo\z@\makeatother
% \section{Interactive code}
% Extracts to |zeckendorf.tex|.
% \label{sec:code:interactive}
%    \begin{macrocode}
\input zeckendorfcore.tex
\xintexprSafeCatcodes

\def\ZeckCmdQ\iftrue{\iffalse}
\let\ZeckCmdX\ZeckCmdQ
\let\ZeckCmdx\ZeckCmdQ
\let\ZeckCmdq\ZeckCmdQ

\newif\ifzeckindices
\def\ZeckCmdL{\zeckindicestrue
              \def\ZeckFromN{\ZeckIndices}\def\ZeckToN{\ZeckNFromIndices}}
\let\ZeckCmdl\ZeckCmdL

\def\ZeckCmdB{\zeckindicesfalse
              \def\ZeckFromN{\ZeckWord}\def\ZeckToN{\ZeckNFromWord}}
\let\ZeckCmdW\ZeckCmdB
\let\ZeckCmdb\ZeckCmdB
\let\ZeckCmdw\ZeckCmdB

\newif\ifzeckfromN
\zeckfromNtrue
\def\ZeckConvert{\csname Zeck\ifzeckfromN From\else To\fi N\endcsname}
\def\ZeckCmdT{\ifzeckfromN\zeckfromNfalse\else\zeckfromNtrue\fi}
\let\ZeckCmdt\ZeckCmdT

\newif\ifzeckmeasuretimes
\expandafter\def\csname ZeckCmd@\endcsname{%
  \ifdefined\xinttheseconds
      \ifzeckmeasuretimes\zeckmeasuretimesfalse\else\zeckmeasuretimestrue\fi
  \else
      \immediate\write128{Sorry, this requires xintexpr 1.4n or later.}%
  \fi
}

\newif\ifzeckevalonly
\def\ZeckCmdE{\ifzeckevalonly\zeckevalonlyfalse\else\zeckevalonlytrue\fi}
\let\ZeckCmde\ZeckCmdE

\ZeckCmdL

\def\ZeckInviteA{commands: (Q)uit, (L)ist, (W)ord, (T)oggle, (E)val-only or @.}

\newlinechar10
\immediate\write128{}
\immediate\write128{Welcome to Zeckendorf 0.9alpha (2025/10/06, JFB).}

\immediate\write128{Command summary (lowercase accepted):^^J
         Q to quit. Also X.^^J
         L for Zeckendorf representations as lists of indices.^^J
         W for Zeckendorf representations as binary words. Also B.^^J
         T to toggle the direction of conversions.^^J
         E to toggle to and from  \string\xintiieval-only mode.^^J
         @ to toggle measurement of execution times.}
\immediate\write128{}
\immediate\write128{%
         Integer input can be an \noexpand\xintiieval expression (but first^^J%
         character must not be a command letter).^^J% Examples:^^J%
         \space\space Examples: 2^100, 1e100, 100!, add(fib(n),n=2..10).^^J%
         \space\space -> fib() function computes Fibonacci numbers.^^J%
         \space\space -> infix operator $ implements Knuth multiplication.^^J%
\space\space\space\space\space E.g. (100$200)$300 or 100$(200$300) or 100$200$300.^^J%$
         List input can (expand to) any comma separated list of integers.^^J%
         \space\space It is also evaluated in \string\xintiieval.^^J%
         \space\space Examples: 2..13 or seq(2^i, i=1, 5, 7).^^J%
         Binary word input is not restricted to be a Zeckendorf word.^^J%
         \space\space It is evaluated in an \string\edef.}
\immediate\write128{}
\immediate\write128{**** empty input is not supported! no linebreaks in input! ****}

\xintloop
\immediate\write128{\ZeckInviteA}
\message{\ifzeckevalonly (eval only mode, hit E to exit it)\else
         \ifzeckfromN Integer -> \ifzeckindices indices\else binary word\fi
         \else
           \ifzeckindices Indices \else Binary word \fi
         -> integer\fi\fi
         : }
%    \end{macrocode}
% Space token at end of |\zeckbuf| is not a problem to us.
% But a |\par| token would be.
%    \begin{macrocode}
\read-1to\zeckbuf
\edef\zeckbuffirst{\expandafter\xintFirstItem\expandafter
    {\detokenize\expandafter{\zeckbuf\relax}}%
}
\ifcsname ZeckCmd\zeckbuffirst\endcsname
  \csname ZeckCmd\zeckbuffirst\expandafter\endcsname
\else
   \ifzeckfromN\edef\ZeckIn{\xintiieval{\zeckbuf}}\else
      \ifzeckindices\edef\ZeckIn{\xintiieval{\zeckbuf}}\else
         \edef\ZeckIn{\zeckbuf}%
      \fi
   \fi
%    \end{macrocode}
% Using the conditional so that this can also be used by default
% with older xint
%    \begin{macrocode}
\ifzeckmeasuretimes\xintresettimer\fi
   \immediate\write128{\ifzeckevalonly\ZeckIn\else\ZeckConvert{\ZeckIn}\fi}%
   \immediate\write128{\ifzeckmeasuretimes
                       \ifzeckevalonly Evaluation \else Conversion \fi
                        took \xinttheseconds s^^J\fi}
\fi
\iftrue
\repeat

\immediate\write128{Bye. Results are also in log file (hard-wrapped too, alas).}
\bye
%    \end{macrocode}
% \catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
% \let</interactive>\relax
% \def<*sty>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</interactive>
%<*sty>
% \makeatletter\global\c@CodelineNo\z@\makeatother
% \section{\LaTeX{} code}
% Extracts to |zeckendorf.sty|.
% \label{sec:code:latex}
%    \begin{macrocode}
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{zeckendorf}
   [2025/10/06 v0.9alpha Zeckendorf representations of big integers (JFB)]%
\RequirePackage{xintexpr}
\input zeckendorfcore.tex
\ZECKrestorecatcodesendinput%
%    \end{macrocode}
% \catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
% \let</sty>\relax
% \def<*dtx>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</sty>
%<*dtx>
% \MakePercentComment
\CheckSum {1309}%
\makeatletter\check@checksum\makeatother%
\Finale%
%% End of file zeckendorf.dtx