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. axiom guess([1, 4, 11, 35, 98, 294, 832, 2401, 6774, 19137, 53466, 148994, 412233], [guessRat], [guessSum, guessProduct], maxLevel==2)
Type: List(Expression(Integer)) 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 axiom guessExpRat [(1+x)^x for x in 0..3]
Type: List(Expression(Integer)) A workaround is necessary, because of bug #128 axiom l := [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)) axiom guessPRec(q)(l, []).1
Type: Expression(Integer) Here are some that are tried: axiom listA := [1,1,2,5,14,42,132]; Type: List(PositiveInteger) axiom listB := [1,2,6,21,80, 322]; Type: List(PositiveInteger) axiom listC := [1,1,2,7,42,429,7436,218348]; Type: List(PositiveInteger) axiom guess(listA, [guessRat], [guessSum, guessProduct])
Type: List(Expression(Integer)) axiom guess(listB, [guessRat], [guessSum, guessProduct])
Type: List(Expression(Integer)) axiom guess(listC, [guessRat], [guessProduct]).1
Type: Expression(Integer) axiom listD := [1,1,2,6,26,162,1450,18626]; Type: List(PositiveInteger) axiom listE := [1,1,2,6,28,202,2252]; Type: List(PositiveInteger) axiom guess(listD, [guessRat], [guessProduct]).1 axiom li := [-86, -975, -100, -1728, -31213]; Type: List(Integer) axiom guess(li, [guessRat], [guessSum, guessProduct])
Type: List(Expression(Integer)) "Most" sequences arising in combinatorics are P-recursive: axiom guessPRec([1,1,6,54,660,10260,194040,4326840,111177360,3234848400,105135861600]).1.function axiom guess([1,1,2,7,40,355,4720,91690,2559980,101724390], [guessRat], [guessSum, guessProduct], maxLevel==2)
Type: List(Expression(Integer)) ... --Thomas, Sun, 27 Jan 2008 04:29:36 -0800 reply axiom guess([1, 2, 3, 7, 11, 16, 26, 36, 56, 81, 131, 183, 287, 417, 677], [guessRat], [guessSum, guessProduct], maxLevel==2)
Type: List(Expression(Integer)) axiom guess([1,1,2,7,40,355,4720,91690,2559980,101724390,5724370860,455400049575], [guessRat], [guessSum, guessProduct], maxLevel==2)
Type: List(Expression(Integer)) axiom guess([1,1,4,35,545,13520,499215,26269200,1917388310,191268774585], [guessRat], [guessSum, guessProduct], maxLevel==2)
Type: List(Expression(Integer)) |

![\label{eq5}\left[{\prod_{
\displaystyle
{{p_{7}}= 0}}^{
\displaystyle
{n - 1}}{{{4 \ {p_{7}}}+ 2}\over{{p_{7}}+ 2}}}\right]
\label{eq5}\left[{\prod_{
\displaystyle
{{p_{7}}= 0}}^{
\displaystyle
{n - 1}}{{{4 \ {p_{7}}}+ 2}\over{{p_{7}}+ 2}}}\right]](images/8493863690807582625-16.0px.png)

![\label{eq10}\left[{\sum_{
\displaystyle
{{s_{15}}= 0}}^{
\displaystyle
{n - 1}}{{\sum_{
\displaystyle
{{s_{14}}= 0}}^{
\displaystyle
{{s_{15}}- 1}}{\left[{f \left({s_{14}}\right)}\mbox{\rm :}{{{\left({{s_{14}}^{10}}-{{71}\ {{s_{14}}^9}}+{{2210}\ {{s_{14}}^8}}-{{395
90}\ {{s_{14}}^7}}+{{450233}\ {{s_{14}}^6}}-{{3379103}\ {{s_{1
4}}^5}}+{{16834180}\ {{s_{14}}^4}}-{{54439860}\ {{s_{14}}^3}}+{{107786016}\ {{s_{14}}^2}}-{{115114176}\ {s_{14}}}+{47900160}\right)}\ {f \left({s_{14}}\right)}}= 0}\right]}}+ 1}}+ 1 \right]
\label{eq10}\left[{\sum_{
\displaystyle
{{s_{15}}= 0}}^{
\displaystyle
{n - 1}}{{\sum_{
\displaystyle
{{s_{14}}= 0}}^{
\displaystyle
{{s_{15}}- 1}}{\left[{f \left({s_{14}}\right)}\mbox{\rm :}{{{\left({{s_{14}}^{10}}-{{71}\ {{s_{14}}^9}}+{{2210}\ {{s_{14}}^8}}-{{395
90}\ {{s_{14}}^7}}+{{450233}\ {{s_{14}}^6}}-{{3379103}\ {{s_{1
4}}^5}}+{{16834180}\ {{s_{14}}^4}}-{{54439860}\ {{s_{14}}^3}}+{{107786016}\ {{s_{14}}^2}}-{{115114176}\ {s_{14}}}+{47900160}\right)}\ {f \left({s_{14}}\right)}}= 0}\right]}}+ 1}}+ 1 \right]](images/8414237468376413726-16.0px.png)