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# Edit detail for SandBox.GuessingSequence revision 20 of 491

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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) (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 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] (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)] (3)
Type: List(Polynomial(Integer))
axiom
guessPRec(q)(l, []).1 (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]) (5)
Type: List(Expression(Integer))
axiom
guess(listB, [guessRat], [guessSum, guessProduct]) (6)
Type: List(Expression(Integer))
axiom
guess(listC, [guessRat], [guessProduct]).1 (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]) (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] (9)
Type: List(PositiveInteger?)
axiom
guessPRec(l).1 (10)
Type: Expression(Integer)

axiom
guess([1,1,2,7,40,355,4720,91690,2559980,101724390], [guessRat], [guessSum, guessProduct], maxLevel==2) (11)
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) (12)
Type: List(Expression(Integer))

axiom
guess([1,1,2,7,40,355,4720,91690,2559980,101724390,5724370860,455400049575], [guessRat], [guessSum, guessProduct], maxLevel==2) (13)
Type: List(Expression(Integer))

axiom
guess([1,1,4,35,545,13520,499215,26269200,1917388310,191268774585], [guessRat], [guessSum, guessProduct], maxLevel==2) (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) (15)
Type: List(Expression(Integer))