MathAction SandBox.GuessingSequence

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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)

$\label{eq1}\left[ \right]$

(1)

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 $\prod_{i=0}^nq(i)$ 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...

axiom

guessExpRat [(1+x)^x for x in 0..3]

$\label{eq2}\left[{{\left(n + 1 \right)}^n}\right]$

(2)

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)]

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[ 1, \: 1, \:{q + 1}, \:{{q^2}+ q + 1}, \:{{q^4}+{q^3}+{q^2}+ q + 1}, \: \right. \ \ \displaystyle \left.{{q^6}+{q^5}+{2 \ {q^4}}+{q^3}+{q^2}+ q + 1}, \: \right. \ \ \displaystyle \left.{{q^9}+{q^8}+{q^7}+{2 \ {q^6}}+{2 \ {q^5}}+{2 \ {q^4}}+{q^3}+{q^2}+ q + 1}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle {q^{12}}+{q^{11}}+{2 \ {q^{10}}}+{2 \ {q^9}}+{2 \ {q^8}}+{2 \ {q^7}}+{3 \ {q^6}}+{2 \ {q^5}}+{2 \ {q^4}}+ \ \ \displaystyle {q^3}+{q^2}+ q + 1$

(3)

Type: List(Polynomial(Integer))

axiom

guessPRec(q)(l, []).1

$\label{eq4}\left[{f \left({n}\right)}\mbox{\rm :}{{{{q \ {f \left({n}\right)}\ {q^n}}-{f \left({n + 2}\right)}+{f \left({n + 1}\right)}}= 0}, \:{{f \left({0}\right)}= 1}, \:{{f \left({1}\right)}= 1}}\right]$

(4)

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])

$\label{eq5}\left[{\prod_{ \displaystyle {{p_{7}}= 0}}^{ \displaystyle {n - 1}}{{{4 \ {p_{7}}}+ 2}\over{{p_{7}}+ 2}}}\right]$

(5)

Type: List(Expression(Integer))

axiom

guess(listB, [guessRat], [guessSum, guessProduct])

$\label{eq6}\left[ \right]$

(6)

Type: List(Expression(Integer))

axiom

guess(listC, [guessRat], [guessProduct]).1

$\label{eq7}\prod_{ \displaystyle {{p_{8}}= 0}}^{ \displaystyle {n - 1}}{\prod_{ \displaystyle {{p_{7}}= 0}}^{ \displaystyle {{p_{8}}- 1}}{{{{27}\ {{p_{7}}^2}}+{{54}\ {p_{7}}}+{24}}\over{{{1 6}\ {{p_{7}}^2}}+{{32}\ {p_{7}}}+{12}}}}$

(7)

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];

Type: List(Integer)

axiom

guess(li, [guessRat], [guessSum, guessProduct])

$\label{eq8}\left[ \right]$

(8)

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)

$\label{eq9}\left[ \right]$

(9)

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)

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

(10)

Type: List(Expression(Integer))

... --Thomas, Wed, 30 Jan 2008 12:12:37 -0800 reply

axiom

guess([1,1,2,7,40,355,4720,91690,2559980,101724390,5724370860,455400049575], [guessRat],
[guessSum, guessProduct], maxLevel==2)

$\label{eq11}\left[ \right]$

(11)

Type: List(Expression(Integer))

... --Thomas, Wed, 30 Jan 2008 12:14:08 -0800 reply

axiom

guess([1,1,4,35,545,13520,499215,26269200,1917388310,191268774585], [guessRat], [guessSum,
guessProduct], maxLevel==2)

$\label{eq12}\left[ \right]$

(12)

Type: List(Expression(Integer))

Subject: Be Bold !!

	( 7 subscribers )