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SandBox.GuessingSequence

last edited 8 months ago by beccari

Edit detail for SandBox.GuessingSequence revision 20 of 598

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Editor: test1 Time: 2013/03/14 19:51:54 GMT-7
Note:

changed:
-guessPRec([1,1,6,54,660,10260,194040,4326840,111177360,3234848400,105135861600]).1.function
l := [1,1,6,54,660,10260,194040,4326840,111177360,3234848400,105135861600,3775206204000]
guessPRec(l).1

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

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

(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

l := [1,1,6,54,660,10260,194040,4326840,111177360,3234848400,105135861600,3775206204000]

$\label{eq9}\begin{array}{@{}l} \displaystyle \left[ 1, \: 1, \: 6, \:{54}, \:{660}, \:{10260}, \:{194040}, \:{4326840}, \:{111177360}, \: \right. \ \ \displaystyle \left.{3234848400}, \:{105135861600}, \:{3775206204000}\right]$

(9)

Type: List(PositiveInteger?)

axiom

guessPRec(l).1

$\label{eq10}\begin{array}{@{}l} \displaystyle \left[{{{f \left({n}\right)}\mbox{\rm :}}{{{f \left({n + 2}\right)}+{{\left(-{4 \ n}- 6 \right)}\ {f \left({n + 1}\right)}}+{{\left({2 \ {{n}^{2}}}+{4 \ n}\right)}\ {f \left({n}\right)}}}= 0}}, \right. \ \ \displaystyle \left.\:{{f \left({0}\right)}= 1}, \:{{f \left({1}\right)}= 1}\right]$

(10)

Type: Expression(Integer)

axiom

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

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

(11)

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)

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

(12)

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)

$\label{eq13}\left[ \right]$

(13)

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)

$\label{eq14}\left[ \right]$

(14)

Type: List(Expression(Integer))

axiom

x1 := -4;

Type: Integer

axiom

x2 := -23/18;

Type: Fraction(Integer)

axiom

x3 := -139/225;

Type: Fraction(Integer)

axiom

x4 := -191833/529200;

Type: Fraction(Integer)

axiom

x5 := -472217/1984500;

Type: Fraction(Integer)

axiom

x6 := -48425779/288149400;

Type: Fraction(Integer)

axiom

x7 := -106497287263/852201850500;

Type: Fraction(Integer)

axiom

x8 := -25074629843/259718659200;

Type: Fraction(Integer)

axiom

x9 := -2162241552187/28147009690800;

Type: Fraction(Integer)

axiom

x10 := -2967138724292741/47418328992434400;

Type: Fraction(Integer)

axiom

x11 := -129037676381827/2483817232937040;

Type: Fraction(Integer)

axiom

x12 := -1570296205027456889/35834708624282568000;

Type: Fraction(Integer)

axiom

x13 := -196315863027088338517/5240826136301325570000;

Type: Fraction(Integer)

axiom

x14 := -182798242115965865171/5643966608324504460000;

Type: Fraction(Integer)

axiom

x15 := -143828683113808323224449/5085617054572401697350000;

Type: Fraction(Integer)

axiom

x16 := -17140536169050680284163795011/688128740913726186786391680000;

Type: Fraction(Integer)

axiom

x17 := -17141645969372168004324275011/775448107658460380942998200000;

Type: Fraction(Integer)

axiom

x18 := -463312933007625360900074503/23458934349331574549536080000;

Type: Fraction(Integer)

axiom

x19 := -23469273048929307035152061800459/1322079072947671102150629783240000;

Type: Fraction(Integer)

axiom

x20 := -572443896207988534144011140099/35683645693594361731460992800000;

Type: Fraction(Integer)

axiom

guess([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20], [guessRat], [guessSum, guessProduct], maxLevel==8)

$\label{eq15}\left[ \right]$

(15)

Type: List(Expression(Integer))