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last edited 8 months ago by beccari |
Edit detail for SandBox.GuessingSequence revision 13 of 598
added: listC := [1,1,2,7,42,429,7436,218348]; added: guess(listC, [guessRat], [guessProduct])
This page makes test uses of the guessing package by Martin Rubey. Feel free to add new sequences or change the sequences to ones you like to try.See GuessingFormulasForSequences for some explanations.
\begin{equation*} \label{eq1}\left[ \right]?\end{equation*}axiomguess([1, 4, 11, 35, 98, 294, 832, 2401, 6774, 19137, 53466, 148994, 412233], [guessRat], [guessSum, guessProduct], maxLevel==2)Type: List Record(function: Expression Integer,order: NonNegativeInteger?)The answer being an empty list tells us, that there is no rational function of total degree less than 13, that generates these numbers. Furthermore, for $q$ being such a rational function, there is no formula of the form $\prod_{i=0}^nq(i)$ or $\sum_{i=0}^nq(i)$, nor $\prod_{i_1=0}^n\prod_{i_2=0}^{i_1}q(i_2)$, nor replacing the products by sums. In fact, if you look at Sloane's encyclopedia, you will find a good reason for that: I'd by very surprised to find such a simple formula for such a family of objects...
\begin{equation*} \label{eq2}\left[{\left[{function ={{\left(n + 1 \right)}^n}}, \:{order = 0}\right]?}\right]\end{equation*}axiomguessExpRat [(1+x)^x for x in 0..3]Type: List Record(function: Expression Integer,order: NonNegativeInteger?)A workaround is necessary, because of bug #128
\begin{equation*} \label{eq3}\begin{array}{@{}l} \displaystyle \left[ 1, \: 1, \:{q + 1}, \:{{q^2}+ q + 1}, \:{{q^4}+{q^3}+{q^2}+ q + 1}, \: \right. \ \ \displaystyle \left.{{q^6}+{q^5}+{2 \ {q^4}}+{q^3}+{q^2}+ q + 1}, \: \right. \ \ \displaystyle \left.{{q^9}+{q^8}+{q^7}+{2 \ {q^6}}+{2 \ {q^5}}+{2 \ {q^4}}+{q^3}+{q^2}+ q + 1}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle {q^{12}}+{q^{11}}+{2 \ {q^{10}}}+{2 \ {q^9}}+{2 \ {q^8}}+{2 \ {q^7}}+{3 \ {q^6}}+{2 \ {q^5}}+{2 \ {q^4}}+{q^3}+ \ \ \displaystyle {q^2}+ q + 1 \end{array} }, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle {q^{16}}+{q^{15}}+{q^{14}}+{2 \ {q^{13}}}+{3 \ {q^{12}}}+{3 \ {q^{11}}}+{3 \ {q^{10}}}+{3 \ {q^9}}+{3 \ {q^8}}+ \ \ \displaystyle {3 \ {q^7}}+{3 \ {q^6}}+{2 \ {q^5}}+{2 \ {q^4}}+{q^3}+{q^2}+ q + 1 \end{array} }\right] \end{array} \end{equation*}axioml := [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)]Type: List Polynomial Integer\begin{equation*} \label{eq4}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle function = \ \ \displaystyle {\left[{f \left({n}\right)}\mbox{\rm :}{{{-{q \ {f \left({n}\right)}\ {q^n}}+{f \left({n + 2}\right)}-{f \left({n + 1}\right)}}= 0}, \:{{f \left({0}\right)}= 1}, \:{{f \left({1}\right)}= 1}}\right]?} \end{array} }, \:{order = 0}\right] \end{array} \end{equation*}axiomguessPRec(q)(l, []).1Type: Record(function: Expression Integer,order: NonNegativeInteger?)Here are some that are tried:
axiomlistA := [1,1,2,5,14,42,132];Type: List PositiveInteger?axiomlistB := [1,2,6,21,80, 322];Type: List PositiveInteger?axiomlistC := [1,1,2,7,42,429,7436,218348];Type: List PositiveInteger?\begin{equation*} \label{eq5}\left[{\left[{function ={\prod_{ \displaystyle {{p_{7}}= 0}}^{ \displaystyle {n - 1}}{{{4 \ {p_{7}}}+ 2}\over{{p_{7}}+ 2}}}}, \:{order = 0}\right]}\right]\end{equation*}axiomguess(listA, [guessRat], [guessSum, guessProduct])Type: List Record(function: Expression Integer,order: NonNegativeInteger?)\begin{equation*} \label{eq6}\left[ \right]?\end{equation*}axiomguess(listB, [guessRat], [guessSum, guessProduct])Type: List Record(function: Expression Integer,order: NonNegativeInteger?)\begin{equation*} \label{eq7}\begin{array}{@{}l} \displaystyle \left[ \left[{function ={\prod_{ \displaystyle {{p_{8}}= 0}}^{ \displaystyle {n - 1}}{\prod_{ \displaystyle {{p_{7}}= 0}}^{ \displaystyle {{p_{8}}- 1}}{{{{27}\ {{p_{7}}^2}}+{{54}\ {p_{7}}}+{24}}\over{{{1 6}\ {{p_{7}}^2}}+{{32}\ {p_{7}}}+{12}}}}}}, \: \right. \ \ \displaystyle \left.{order = 0}\right] \right] \end{array} \end{equation*}axiomguess(listC, [guessRat], [guessProduct])Type: List Record(function: Expression Integer,order: NonNegativeInteger?)axiomli := [-86, -975, -100, -1728, -31213];Type: List Integer\begin{equation*} \label{eq8}\left[ \right]?\end{equation*}axiomguess(li, [guessRat], [guessSum, guessProduct])Type: List Record(function: Expression Integer,order: NonNegativeInteger?)"Most" sequences arising in combinatorics are P-recursive:
\begin{equation*} \label{eq9}\left[{f \left({n}\right)}\mbox{\rm :}{{{-{f \left({n + 2}\right)}+{{\left({4 \ n}+ 6 \right)}\ {f \left({n + 1}\right)}}+{{\left(-{2 \ {n^2}}-{4 \ n}\right)}\ {f \left({n}\right)}}}= 0}, \:{{f \left({0}\right)}= 1}, \:{{f \left({1}\right)}= 1}}\right]?\end{equation*}axiomguessPRec([1,1,6,54,660,10260,194040,4326840,111177360,3234848400,105135861600]).1.functionType: Expression Integer\begin{equation*} \label{eq10}\left[ \right]?\end{equation*}axiomguess([1,1,2,7,40,355,4720,91690,2559980,101724390], [guessRat], [guessSum, guessProduct], maxLevel==2)Type: List Record(function: Expression Integer,order: NonNegativeInteger?)... --Thomas, Sun, 27 Jan 2008 04:29:36 -0800 reply\begin{equation*} \label{eq11}\left[ \right]?\end{equation*}axiomguess([1, 2, 3, 7, 11, 16, 26, 36, 56, 81, 131, 183, 287, 417, 677], [guessRat], [guessSum, guessProduct], maxLevel==2)Type: List Record(function: Expression Integer,order: NonNegativeInteger?)\begin{equation*} \label{eq12}\left[ \right]?\end{equation*}axiomguess([1,1,2,7,40,355,4720,91690,2559980,101724390,5724370860,455400049575], [guessRat], [guessSum, guessProduct], maxLevel==2)Type: List Record(function: Expression Integer,order: NonNegativeInteger?)\begin{equation*} \label{eq13}\left[ \right]?\end{equation*}axiomguess([1,1,4,35,545,13520,499215,26269200,1917388310,191268774585], [guessRat], [guessSum, guessProduct], maxLevel==2)Type: List Record(function: Expression Integer,order: NonNegativeInteger?)
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