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 
being such a rational
function, there is no formula of the form 
or
, nor 
,
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...
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
>> Error detected within library code:
index out of range
axiom
li := [-86, -975, -100, -1728, -31213];
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
>> Error detected within library code:
index out of range
axiom
guess([1,1,2,7,40,355,4720,91690,2559980,101724390], [guessRat], [guessSum, guessProduct],
maxLevel==2)
Type: List(Expression(Integer))
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{eq1}\left[ \right]](images/5593850768678758914-16.0px.png "\label{eq1}\left[ \right]")
![\label{eq2}\left[{{\left(n + 1 \right)}^n}\right]](images/5269408325183924137-16.0px.png "\label{eq2}\left[{{\left(n + 1 \right)}^n}\right]")
![\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}}+
\
\
\displaystyle
{q^3}+{q^2}+ q + 1](images/2969832125479833061-16.0px.png "\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}}+
\
\
\displaystyle
{q^3}+{q^2}+ q + 1")
![\label{eq4}\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]](images/1760057335759037990-16.0px.png "\label{eq4}\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]")
![\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{eq5}\left[{\prod_{
\displaystyle
{{p_{7}}= 0}}^{
\displaystyle
{n - 1}}{{{4 \ {p_{7}}}+ 2}\over{{p_{7}}+ 2}}}\right]")
![\label{eq6}\left[ \right]](images/3015632402752253779-16.0px.png "\label{eq6}\left[ \right]")
![\label{eq8}\left[ \right]](images/1204202700703939827-16.0px.png "\label{eq8}\left[ \right]")
![\label{eq9}\left[ \right]](images/8299985760225880582-16.0px.png "\label{eq9}\left[ \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 "\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{eq11}\left[ \right]](images/8713422998239773744-16.0px.png "\label{eq11}\left[ \right]")
![\label{eq12}\left[ \right]](images/2574718782038032715-16.0px.png "\label{eq12}\left[ \right]")