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## [Axiom-developer] [Guessing formulas for sequences]

 From: kratt6 Subject: [Axiom-developer] [Guessing formulas for sequences] Date: Sun, 20 Mar 2005 12:08:29 -0600

Changes
http://page.axiom-developer.org/zope/mathaction/GuessingFormulasForSequences/diff
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??changed:
-We present a software package that guesses formulas for sequences of rational
Important Note

There are two bugs in the version of Axiom currently run on this server that
prevent
parts of the package from working properly. One bug, #63, prevents 'guessPade'
from
working. Another bug, #8 messes up the output by missing some parenthesis.
Preliminary
patches for both are available, we hope that they will soon be fixed here, too.

Please add other bugs you find to IssueTracker by clicking on "issues" on the
top
left of any page and filling out the appropriate forms.

Finally, please feel free to try this package in the SandBox!

Abstract

We present a software package that guesses formulas for sequences of rational

??changed:
-with the 'Axiom' package 'Guess'. The package provides currently four
primitive guessing functions, namely:
with the 'Axiom' package 'Guess'. To load the package we type
\begin{axiom}
)lib RINTERPA RINTERP PCDEN GUESS GUESSINT GUESSP
\end{axiom}
The package provides currently four primitive guessing functions, namely:

??changed:
-Sequence (2) satisfies a simple recurrence, we type::
-
-      guessPade(n, [1, 1, 2, 3, 5], n+->n)\$GuessInteger - -to obtain an answer like -\begin{equation*} - \left[ -{\left[ {function={coefficient -\left( -{-{1 \over {{ x \sp 2}+ x -1}}, \: x, \: n} -\right)}}, -\: {order=1} -\right]} -\right] -\end{equation*} Sequence (2) satisfies a simple recurrence, we type \begin{axiom} guessPade(n, [1, 1, 2, 3, 5], n+->n)$GuessInteger
\end{axiom}

??changed:
-  guessRat(n, [3, 4, 7/2, 18/5, 11/3, 26/7], n+->n)\$GuessInteger - -returns -\begin{equation*} -\left[ -{\left[ {function={{4n+2} \over {n+1}}}, \: {order=3} -\right]} -\right]. -\end{equation*} \begin{axiom} guessRat(n, [3, 4, 7/2, 18/5, 11/3, 26/7], n+->n)$GuessInteger
\end{axiom}

??changed:
-the appropriate function is::
-
-  guess(n, [0, 1, 3, 9, 33], n+->n, [guessRat, guessPade],
-        [guessSum, guessProduct, guessOne])\$GuessInteger - -and the output will be - apart from some information on the progress - -\begin{equation*} -\left[ -{\left[ {function={\sum \sb{\displaystyle {{s \sb {5}}=1}} \sp{\displaystyle -{n -1}} {\prod \sb{\displaystyle {{p \sb {4}}=1}} \sp{\displaystyle {{s \sb -{5}} -1}} {({p \sb {4}}+1)}}}}, \: {order=1} -\right]} -\right]. -\end{equation*} the appropriate function is \begin{axiom} guess(n, [0, 1, 3, 9, 33], n+->n, 2, [guessRat], [guessSum, guessProduct, guessOne])$GuessInteger
\end{axiom}

??changed:
-For example::
-
-      guess(n, [1, 1+q, 1+q+q^2, 1+q+q^2+q^3], n+->n, [guessRat],
-            [guessSum, guessProduct])\$GuessPolynomial - -returns -\begin{equation*} -\left[ -{\left[ {function={\sum \sb{\displaystyle {{s \sb {2}}=1}} -\sp{\displaystyle {n -1}} {q \ {q \sp {\left( {s \sb {2}} -1 -\right)}}}+1}}, -\: {order=1} -\right]} -\right]. -\end{equation*} For example \begin{axiom} guess(n, [1, 1+q, 1+q+q^2, 1+q+q^2+q^3], n+->n, [guessRat], [guessSum, guessProduct])$GuessPolynomial
\end{axiom}

??changed:
-'n+->q^n'. For example::
-
-  guessRat(n, [1,1+q+q^2,(1+q+q^2)*(1+q^2),
-               (1+q^2)*(1+q+q^2+q^3+q^4)], n+->q^n)\$GuessPolynomial - -returns -\begin{equation*} -\left[ -{\left[ {function={{{q \ {q \sp {\left( 2 \ n -\right)}}}+{{\left( --q -1 -\right)} -\ {q \sp n}}+1} \over {{q \sp 3} -{q \sp 2} -q+1}}}, \: {order=1} -\right]} -\right] -\end{equation*} 'n+->q^n'. For example \begin{axiom} guessRat(n, [1,1+q+q^2,(1+q+q^2)*(1+q^2), (1+q^2)*(1+q+q^2+q^3+q^4)], n+->q^n)$GuessPolynomial
\end{axiom}

??changed:
-Sequence (4) ::
-
-  guessPRec(n,[1, 1, 1+q, 1+q+q^2, 1+q+q^2+q^3+q^4,
-             1+q+q^2+q^3+2*q^4+q^5+q^6,
-             1+q+q^2+q^3+2*q^4+2*q^5+2*q^6+q^7+q^8+q^9,
-             (1+q^4+q^6)*(1+q+q^2+q^3+q^4+q^5+q^6),
-             (1+q^4)*(1+q+q^2+q^3+q^4+q^5+2*q^6+2*q^7
-             +2*q^8+2*q^9+q^10+q^11+q^12)], n+->q^n)\$GuessPolynomial - -yields -\begin{equation*} - \left[ -{\left[ {function={rootof -\left( -{{-{{ x -\left( -{n} -\right)} -\ {q \sp n}}+{ x -[11 more lines...] Sequence (4) \begin{axiom} guessPRec(n,[1, 1, 1+q, 1+q+q^2, 1+q+q^2+q^3+q^4, 1+q+q^2+q^3+2*q^4+q^5+q^6, 1+q+q^2+q^3+2*q^4+2*q^5+2*q^6+q^7+q^8+q^9, 1+q+q^2+q^3+q^4+q^5+q^6, (1+q^4)*(1+q+q^2+q^3+q^4+q^5+2*q^6+2*q^7+2*q^8+2*q^9+q^10+q^11+q^12)], n+->q^n)$GuessPolynomial
\end{axiom}

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