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last edited 8 months ago by beccari |
Edit detail for SandBox.GuessingSequence revision 21 of 598
added: \begin{axiom} l := [-1/3,-11/25,-23/49,-13/27,-59/121,-83/169] guess(l, [guessRat], [guessSum, guessProduct], maxLevel==2) \end{axiom} changed: -guess(listD, [guessRat], [guessProduct]).1 -guess(listE, [guessRat], [guessProduct]).1 guess(listD, [guessRat], [guessProduct]) guess(listE, [guessRat], [guessProduct]) added: Power of a P-recursive sequence is again a P-recursive sequence (we switch to text output for the next two sequences because TeX messes the formulas): \begin{axiom} )set output tex off )set output algebra on l := [hermiteH(n, 3)^4 for n in 0..110]; guessPRec(l, safety==10) \end{axiom} We can guess also equation for sequence of polynomials: \begin{axiom} l := [hermiteH(n, x)^4 for n in 0..110]; guessPRec(l, safety==10) )set output tex on )set output algebra off \end{axiom}
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]](images/5593850768678758914-16.0px.png "\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 
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]
![\label{eq2}\left[{{\left(n + 1 \right)}^{n}}\right]](images/6683871585640527075-16.0px.png "\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](images/4636869417498219116-16.0px.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") | (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]](images/1166922280319524024-16.0px.png "\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]](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]") | (5) |
Type: List(Expression(Integer))
axiom
guess(listB,[guessRat], [guessSum, guessProduct])
![\label{eq6}\left[ \right]](images/3015632402752253779-16.0px.png "\label{eq6}\left[ \right]") | (6) |
Type: List(Expression(Integer))
axiom
guess(listC,[guessRat], [guessProduct]).1
 | (7) |
Type: Expression(Integer)
axiom
l := [-1/3,-11/25, -23/49, -13/27, -59/121, -83/169]
![\label{eq8}\left[ -{1 \over 3}, \: -{{11}\over{25}}, \: -{{23}\over{49}}, \: -{{13}\over{27}}, \: -{{59}\over{121}}, \: -{{83}\over{16
9}}\right]](images/4592720528368633478-16.0px.png "\label{eq8}\left[ -{1 \over 3}, \: -{{11}\over{25}}, \: -{{23}\over{49}}, \: -{{13}\over{27}}, \: -{{59}\over{121}}, \: -{{83}\over{16
9}}\right]") | (8) |
Type: List(Fraction(Integer))
axiom
guess(l,[guessRat], [guessSum, guessProduct], maxLevel==2)
![\label{eq9}\left[{{-{2 \ {{n}^{2}}}-{6 \ n}- 3}\over{{4 \ {{n}^{2}}}+{{1
2}\ n}+ 9}}\right]](images/7606205255704515682-16.0px.png "\label{eq9}\left[{{-{2 \ {{n}^{2}}}-{6 \ n}- 3}\over{{4 \ {{n}^{2}}}+{{1
2}\ n}+ 9}}\right]") | (9) |
Type: List(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])
![\label{eq10}\left[ \right]](images/652275695860962513-16.0px.png "\label{eq10}\left[ \right]") | (10) |
Type: List(Expression(Integer))
axiom
guess(listE,[guessRat], [guessProduct])
![\label{eq11}\left[ \right]](images/8713422998239773744-16.0px.png "\label{eq11}\left[ \right]") | (11) |
Type: List(Expression(Integer))
axiom
li := [-86,-975, -100, -1728, -31213];
Type: List(Integer)
axiom
guess(li,[guessRat], [guessSum, guessProduct])
![\label{eq12}\left[ \right]](images/2574718782038032715-16.0px.png "\label{eq12}\left[ \right]") | (12) |
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{eq13}\begin{array}{@{}l}
\displaystyle
\left[ 1, \: 1, \: 6, \:{54}, \:{660}, \:{10260}, \:{194040}, \:{4326840}, \:{111177360}, \: \right.
\
\
\displaystyle
\left.{3234848400}, \:{105135861600}, \:{3775206204000}\right]](images/5170788584070598217-16.0px.png "\label{eq13}\begin{array}{@{}l}
\displaystyle
\left[ 1, \: 1, \: 6, \:{54}, \:{660}, \:{10260}, \:{194040}, \:{4326840}, \:{111177360}, \: \right.
\
\
\displaystyle
\left.{3234848400}, \:{105135861600}, \:{3775206204000}\right]") | (13) |
Type: List(PositiveInteger?)
axiom
guessPRec(l).1
![\label{eq14}\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]](images/4751471392109749243-16.0px.png "\label{eq14}\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]") | (14) |
Type: Expression(Integer)
Power of a P-recursive sequence is again a P-recursive sequence (we switch to text output for the next two sequences because TeX? messes the formulas):
axiom
)set output tex off
axiom
)set output algebra on
l := [hermiteH(n,3)^4 for n in 0..110];
Type: List(Integer)
axiom
guessPRec(l,safety==10)
(22) [ [ f(n): 3 2 (- 2n + 75n - 883n + 3120)f(n + 5) + 5 4 3 2 (8n - 668n + 18104n - 207512n + 1002128n - 1675520)f(n + 4) + 7 6 5 4 3 2 64n - 5440n + 152048n - 1798384n + 7875968n + 9706784n + - 165360320n + 326726400 * f(n + 3) + 9 8 7 6 5 - 256n + 20736n - 513216n + 4234176n + 5618496n + 4 3 2 - 183387456n + 187255936n + 2614783104n - 2278264320n + - 13442457600 * f(n + 2) + 11 10 9 8 7 - 512n + 35072n - 560128n - 600320n + 44719104n + 6 5 4 3 59460096n - 1210722304n - 3830374400n + 5429444608n + 2 41504940032n + 67395551232n + 36498309120 * f(n + 1) + 13 12 11 10 9 2048n - 29696n - 297984n + 2210816n + 28607488n + 8 7 6 5 65614848n - 352019456n - 2713947136n - 8308030464n + 4 3 2 - 14724902912n - 16198819840n - 10931650560n - 4153393152n + - 681246720 * f(n) = 0 ,f(0)= 1, f(1)= 1296, f(2)= 1336336, f(3)= 1049760000] ]
Type: List(Expression(Integer))
We can guess also equation for sequence of polynomials:
axiom
l := [hermiteH(n,x)^4 for n in 0..110];
Type: List(Polynomial(Integer))
axiom
guessPRec(l,safety==10)
(24) [ [ f(n): 6 4 2 2 3 2 8x + (- 16n - 40)x + (10n + 50n + 62)x - 2n - 15n - 37n + - 30 * f(n + 5) + 10 8 2 6 - 128x + (448n + 1408)x + (- 576n - 3616n - 5600)x + 3 2 4 (336n + 3152n + 9696n + 9760)x + 4 3 2 2 5 4 3 (- 88n - 1096n - 5016n - 9968n - 7232)x + 8n + 124n + 752n + 2 2224n + 3200n + 1792 * f(n + 4) + 12 2 10 (1024n + 4096)x + (- 4608n - 32768n - 57344)x + 3 2 8 (8192n + 83968n + 284160n + 317440)x + 4 3 2 6 (- 7296n - 97920n - 488960n - 1076224n - 880640)x + 5 4 3 2 4 (3392n + 56384n + 372096n + 1218048n + 1976896n + 1272064)x + 6 5 4 3 2 - 768n - 15232n - 124896n - 541664n - 1309792n + - 1673472n - 882176 * 2 x + 7 6 5 4 3 2 64n + 1472n + 14384n + 77360n + 247136n + 468608n + 487936n + 215040 * f(n + 3) + 3 2 12 (- 4096n - 40960n - 135168n - 147456)x + 4 3 2 10 (18432n + 237568n + 1140736n + 2420736n + 1916928)x + 5 4 3 2 - 32768n - 516096n - 3233792n - 10080256n - 15636480n + - 9658368 * 8 x + 6 5 4 3 2 29184n + 542208n + 4175360n + 17060352n + 39012352n + 47339520n + 23814144 * 6 x + 7 6 5 4 3 - 13568n - 290048n - 2642688n - 13302528n - 39951616n + 2 - 71583232n - 70842624n - 29869056 * 4 x + 8 7 6 5 4 3072n + 74240n + 780160n + 4655744n + 17254912n + 3 2 40661888n + 59489152n + 49392384n + 17814528 * 2 x + 9 8 7 6 5 4 - 256n - 6912n - 82368n - 568512n - 2504256n - 7299648n + 3 2 - 14077952n - 17319168n - 12331008n - 3870720 * f(n + 2) + 6 5 4 3 2 8192n + 131072n + 860160n + 2965504n + 5668864n + 5701632n + 2359296 * 10 x + 7 6 5 4 3 - 28672n - 540672n - 4321280n - 18980864n - 49496064n + 2 - 76644352n - 65273856n - 23592960 * 8 x + 8 7 6 5 4 36864n + 800768n + 7542784n + 40245248n + 133052416n + 3 2 279130112n + 362930176n + 267436032n + 85524480 * 6 x + 9 8 7 6 5 - 21504n - 529408n - 5745664n - 36082688n - 144510976n + 4 3 2 - 382810112n - 670806016n - 749903872n - 485376000n + - 138608640 * 4 x + 10 9 8 7 6 5632n + 155136n + 1906688n + 13770240n + 64722432n + 5 4 3 2 206893056n + 455585792n + 682500096n + 665804800n + 382009344n + 97910784 * 2 x + 11 10 9 8 7 - 512n - 15616n - 214528n - 1752320n - 9457152n + 6 5 4 3 - 35414016n - 93908992n - 176377856n - 229990400n + 2 - 198344704n - 101842944n - 23592960 * f(n + 1) + 10 9 8 7 6 - 8192n - 163840n - 1433600n - 7225344n - 23224320n + 5 4 3 2 - 49741824n - 71901184n - 69287936n - 42631168n + - 15138816n - 2359296 * 6 x + 11 10 9 8 7 16384n + 385024n + 4014080n + 24485888n + 97026048n + 6 5 4 3 262053888n + 491995136n + 641884160n + 570277888n + 2 328695808n + 110690304n + 16515072 * 4 x + 12 11 10 9 - 10240n - 276480n - 3350528n - 24074240n + 8 7 6 5 - 114114560n - 375576576n - 879288320n - 1474308096n + 4 3 2 - 1756051456n - 1448587264n - 785539072n - 251510784n + - 35979264 * 2 x + 13 12 11 10 9 2048n + 62464n + 863232n + 7150592n + 39574528n + 8 7 6 5 154383360n + 436335616n + 903812096n + 1370926080n + 4 3 2 1502906368n + 1156861952n + 591962112n + 180486144n + 24772608 * f(n) = 0 ,4 8 6 4 2 f(0)= 1, f(1)= 16x , f(2)= 256x - 512x + 384x - 128x + 16, 12 10 8 6 4 f(3)= 4096x - 24576x + 55296x - 55296x + 20736x ] ]
Type: List(Expression(Integer))
axiom
)set output tex on
axiom
)set output algebra off
axiom
guess([1,1, 2, 7, 40, 355, 4720, 91690, 2559980, 101724390], [guessRat], [guessSum, guessProduct], maxLevel==2)
![\label{eq15}\left[ \right]](images/1937748106585371060-16.0px.png "\label{eq15}\left[ \right]") | (15) |
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{eq16}\left[ \right]](images/3763945104387423615-16.0px.png "\label{eq16}\left[ \right]") | (16) |
Type: List(Expression(Integer))
axiom
guess([1,1, 2, 7, 40, 355, 4720, 91690, 2559980, 101724390, 5724370860, 455400049575], [guessRat], [guessSum, guessProduct], maxLevel==2)
![\label{eq17}\left[ \right]](images/85015405605998390-16.0px.png "\label{eq17}\left[ \right]") | (17) |
Type: List(Expression(Integer))
axiom
guess([1,1, 4, 35, 545, 13520, 499215, 26269200, 1917388310, 191268774585], [guessRat], [guessSum, guessProduct], maxLevel==2)
![\label{eq18}\left[ \right]](images/3479779599092591769-16.0px.png "\label{eq18}\left[ \right]") | (18) |
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{eq19}\left[ \right]](images/5764799739952902792-16.0px.png "\label{eq19}\left[ \right]") | (19) |
Type: List(Expression(Integer))