Edit detail for SandBox.GuessingSequence revision 43 of 598

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.
fricas
guess([1, 4, 11, 35, 98, 294, 832, 2401, 6774, 19137, 53466, 148994, 412233], [guessRat], [guessSum, guessProduct], maxLevel==2)
\begin{equation*}
\label{eq1}\left[ \right]?\end{equation*}
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 $q$ 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...
fricas
guessExpRat [(1+x)^x for x in 0..3]
\begin{equation*}
\label{eq2}\left[{{\left(n + 1 \right)}^{n}}\right]?\end{equation*}
Type: List(Expression(Integer))

A workaround is necessary, because of bug #128
fricas
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)]
\begin{equation*}
\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 
\end{array}
}, \right.
\
\
\displaystyle
\left.\: \right.
\
\
\displaystyle
\left.{
\begin{array}{@{}l}
\displaystyle
{{q}^{16}}+{{q}^{15}}+{{q}^{14}}+{2 \ {{q}^{13}}}+{3 \ {{q}^{1
2}}}+{3 \ {{q}^{11}}}+{3 \ {{q}^{10}}}+{3 \ {{q}^{9}}}+ 
\
\
\displaystyle
{3 \ {{q}^{8}}}+{3 \ {{q}^{7}}}+{3 \ {{q}^{6}}}+{2 \ {{q}^{5}}}+{2 \ {{q}^{4}}}+{{q}^{3}}+{{q}^{2}}+ q + 1 
\end{array}
}\right] 
\end{array}
\end{equation*}
Type: List(Polynomial(Integer))
fricas
guessPRec(q)(l, []).1
\begin{equation*}
\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] 
\end{array}
\end{equation*}
Type: Expression(Integer)

Here are some that are tried:
fricas
listA := [1,1,2,5,14,42,132];
Type: List(PositiveInteger?)
fricas
listB := [1,2,6,21,80, 322];
Type: List(PositiveInteger?)
fricas
listC := [1,1,2,7,42,429,7436,218348];
Type: List(PositiveInteger?)
fricas
guess(listA, [guessRat], [guessSum, guessProduct])
\begin{equation*}
\label{eq5}\left[{\prod_{
\displaystyle
{{p_{7}}= 0}}^{
\displaystyle
{n - 1}}{{{4 \ {p_{7}}}+ 2}\over{{p_{7}}+ 2}}}\right]\end{equation*}
Type: List(Expression(Integer))
fricas
guess(listB, [guessRat], [guessSum, guessProduct])
\begin{equation*}
\label{eq6}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))
fricas
guess(listC, [guessRat], [guessProduct]).1
\begin{equation}
\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}}}}\end{equation}
Type: Expression(Integer)

fricas
l := [-1/3,-11/25,-23/49,-13/27,-59/121,-83/169]
\begin{equation*}
\label{eq8}\left[ -{1 \over 3}, \: -{{11}\over{25}}, \: -{{23}\over{49}}, \: -{{13}\over{27}}, \: -{{59}\over{121}}, \: -{{83}\over{16
9}}\right]\end{equation*}
Type: List(Fraction(Integer))
fricas
guess(l, [guessRat], [guessSum, guessProduct], maxLevel==2)
\begin{equation*}
\label{eq9}\left[{{-{2 \ {{n}^{2}}}-{6 \  n}- 3}\over{{4 \ {{n}^{2}}}+{{1
2}\  n}+ 9}}\right]\end{equation*}
Type: List(Expression(Integer))

fricas
listD := [1,1,2,6,26,162,1450,18626];
Type: List(PositiveInteger?)
fricas
listE := [1,1,2,6,28,202,2252];
Type: List(PositiveInteger?)
fricas
guess(listD, [guessRat], [guessProduct])
\begin{equation*}
\label{eq10}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))
fricas
guess(listE, [guessRat], [guessProduct])
\begin{equation*}
\label{eq11}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))

fricas
li :=  [-86, -975, -100, -1728, -31213];
Type: List(Integer)
fricas
guess(li, [guessRat], [guessSum, guessProduct])
\begin{equation*}
\label{eq12}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))

"Most" sequences arising in combinatorics are P-recursive:
fricas
l := [1,1,6,54,660,10260,194040,4326840,111177360,3234848400,105135861600,3775206204000]
\begin{equation*}
\label{eq13}\begin{array}{@{}l}
\displaystyle
\left[ 1, \: 1, \: 6, \:{54}, \:{660}, \:{10260}, \:{194040}, \:{4326840}, \:{111177360}, \: \right.
\
\
\displaystyle
\left.{3234848400}, \:{105135861600}, \:{3775206204000}\right] \end{array}
\end{equation*}
Type: List(PositiveInteger?)
fricas
guessPRec(l).1
\begin{equation*}
\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] \end{array}
\end{equation*}
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):
fricas
)set output tex off
 
fricas
)set output algebra on

l := [hermiteH(n, 3)^4 for n in 0..110];
Type: List(Integer)
fricas
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:
fricas
l := [hermiteH(n, x)^4 for n in 0..110];
Type: List(Polynomial(Integer))
fricas
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))
fricas
)set output tex on
 
fricas
)set output algebra off

fricas
guess([1,1,2,7,40,355,4720,91690,2559980,101724390], [guessRat], [guessSum, guessProduct], maxLevel==2)
\begin{equation*}
\label{eq15}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))



... --Thomas,  Sun, 27 Jan 2008 04:29:36 -0800 reply
fricas
guess([1, 2, 3, 7, 11, 16, 26, 36, 56, 81, 131, 183, 287, 417, 677], [guessRat], [guessSum, guessProduct], maxLevel==2)
\begin{equation*}
\label{eq16}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))

... --Thomas,  Wed, 30 Jan 2008 12:12:37 -0800 reply
fricas
guess([1,1,2,7,40,355,4720,91690,2559980,101724390,5724370860,455400049575], [guessRat], [guessSum, guessProduct], maxLevel==2)
\begin{equation*}
\label{eq17}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))

... --Thomas,  Wed, 30 Jan 2008 12:14:08 -0800 reply
fricas
guess([1,1,4,35,545,13520,499215,26269200,1917388310,191268774585], [guessRat], [guessSum, guessProduct], maxLevel==2)
\begin{equation*}
\label{eq18}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))

fricas
x1 := 1;
Type: PositiveInteger?
fricas
x2 := -3/8;
Type: Fraction(Integer)
fricas
x3 := 3/128;
Type: Fraction(Integer)
fricas
x4 := 3/512;
Type: Fraction(Integer)
fricas
x5 := 63/32768;
Type: Fraction(Integer)
fricas
x6 := 27/32768;
Type: Fraction(Integer)
fricas
x7 := 1899/4194304;
Type: Fraction(Integer)
fricas
x8 := 81/262144;
Type: Fraction(Integer)
fricas
x9 := 543483/2147483648;
Type: Fraction(Integer)
fricas
x10 := 32427/134217728;
Type: Fraction(Integer)
fricas
x11 := 72251109/274877906944;
Type: Fraction(Integer)
fricas
x12 := 2752623/8589934592;
Type: Fraction(Integer)
fricas
x13 := 30413055339/70368744177664;
Type: Fraction(Integer)
fricas
x14 := 87745113/137438953472;
Type: Fraction(Integer)
fricas
x15 := 9228545313147/9007199254740992;
Type: Fraction(Integer)
fricas
guess([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10], [guessRat], [guessSum, guessProduct], maxLevel==8)
\begin{equation*}
\label{eq19}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))
fricas
guess([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10])
\begin{equation*}
\label{eq20}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))
fricas
guessPRec([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10])
\begin{equation*}
\label{eq21}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))
fricas
guessADE([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15])
\begin{equation*}
\label{eq22}\begin{array}{@{}l}
\displaystyle
\left[ \left[{{\left[{x}^{n}\right]?}{f \left({x}\right)}\mbox{\rm :}{{{{{x}^{2}}\ {{f_{\ }^{,}}\left({x}\right)}}-{4 \ {{f \left({x}\right)}^{2}}}-{3 \  x \ {f \left({x}\right)}}+ 4}= 0}}, \: \right.
\
\
\displaystyle
\left.{{f \left({x}\right)}={1 -{{3 \  x}\over 8}+{{3 \ {{x}^{2}}}\over{128}}+{{3 \ {{x}^{3}}}\over{512}}+{O \left({{x}^{4}}\right)}}}\right] \right] 
\end{array}
\end{equation*}
Type: List(Expression(Integer))

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