Edit detail for SandBox.GuessingSequence revision 605 of 608

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
(1) -> 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}}{\frac{{4 \ {p_{7}}}+ 2}{{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}}{\frac{{{27}\ {{p_{7}}^{2}}}+{{54}\ {p_{7}}}+{24}}{{{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}\begin{array}{@{}l}
\displaystyle
\left[ -{\frac{1}{3}}, \: -{\frac{11}{25}}, \: -{\frac{23}{49}}, \: -{\frac{13}{27}}, \: -{\frac{59}{121}}, \right.
\
\
\displaystyle
\left.\: -{\frac{83}{169}}\right] 
\end{array}
\end{equation*}
Type: List(Fraction(Integer))
fricas
guess(l, [guessRat], [guessSum, guessProduct], maxLevel==2)
\begin{equation*}
\label{eq9}\left[{\frac{-{2 \ {{n}^{2}}}-{6 \  n}- 3}{{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
             (- 2 n  + 75 n  - 883 n + 3120)f(n + 5)
           + 
                   5        4          3           2
               (8 n  - 668 n  + 18104 n  - 207512 n  + 1002128 n - 1675520)
            *
               f(n + 4)
           + 
                     7         6           5            4            3
                 64 n  - 5440 n  + 152048 n  - 1798384 n  + 7875968 n
               + 
                          2
                 9706784 n  - 165360320 n + 326726400
            *
               f(n + 3)
           + 
                        9          8           7            6            5
                 - 256 n  + 20736 n  - 513216 n  + 4234176 n  + 5618496 n
               + 
                              4              3               2
                 - 183387456 n  + 187255936 n  + 2614783104 n  - 2278264320 n
               + 
                 - 13442457600
            *
               f(n + 2)
           + 
                        11          10           9           8             7
                 - 512 n   + 35072 n   - 560128 n  - 600320 n  + 44719104 n
               + 
                           6               5               4               3
                 59460096 n  - 1210722304 n  - 3830374400 n  + 5429444608 n
               + 
                              2
                 41504940032 n  + 67395551232 n + 36498309120
            *
               f(n + 1)
           + 
                       13          12           11            10             9
                 2048 n   - 29696 n   - 297984 n   + 2210816 n   + 28607488 n
               + 
                           8              7               6               5
                 65614848 n  - 352019456 n  - 2713947136 n  - 8308030464 n
               + 
                                4                3                2
                 - 14724902912 n  - 16198819840 n  - 10931650560 n
               + 
                 - 4153393152 n - 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
                 8 x  + (- 16 n - 40)x  + (10 n  + 50 n + 62)x  - 2 n  - 15 n
               + 
                 - 37 n - 30
            *
               f(n + 5)
           + 
                        10                  8           2                  6
                 - 128 x   + (448 n + 1408)x  + (- 576 n  - 3616 n - 5600)x
               + 
                       3         2                  4
                 (336 n  + 3152 n  + 9696 n + 9760)x
               + 
                        4         3         2                  2      5        4
                 (- 88 n  - 1096 n  - 5016 n  - 9968 n - 7232)x  + 8 n  + 124 n
               + 
                      3         2
                 752 n  + 2224 n  + 3200 n + 1792
            *
               f(n + 4)
           + 
                                 12            2                    10
                 (1024 n + 4096)x   + (- 4608 n  - 32768 n - 57344)x
               + 
                        3          2                      8
                 (8192 n  + 83968 n  + 284160 n + 317440)x
               + 
                          4          3           2                       6
                 (- 7296 n  - 97920 n  - 488960 n  - 1076224 n - 880640)x
               + 
                           5          4           3            2
                     3392 n  + 56384 n  + 372096 n  + 1218048 n  + 1976896 n
                   + 
                     1272064
                *
                    4
                   x
               + 
                            6          5           4           3            2
                     - 768 n  - 15232 n  - 124896 n  - 541664 n  - 1309792 n
                   + 
                     - 1673472 n - 882176
                *
                    2
                   x
               + 
                     7         6          5          4           3           2
                 64 n  + 1472 n  + 14384 n  + 77360 n  + 247136 n  + 468608 n
               + 
                 487936 n + 215040
            *
               f(n + 3)
           + 
                          3          2                      12
                 (- 4096 n  - 40960 n  - 135168 n - 147456)x
               + 
                         4           3            2                        10
                 (18432 n  + 237568 n  + 1140736 n  + 2420736 n + 1916928)x
               + 
                              5           4            3             2
                     - 32768 n  - 516096 n  - 3233792 n  - 10080256 n
                   + 
                     - 15636480 n - 9658368
                *
                    8
                   x
               + 
                            6           5            4             3
                     29184 n  + 542208 n  + 4175360 n  + 17060352 n
                   + 
                               2
                     39012352 n  + 47339520 n + 23814144
                *
                    6
                   x
               + 
                              7           6            5             4
                     - 13568 n  - 290048 n  - 2642688 n  - 13302528 n
                   + 
                                 3             2
                     - 39951616 n  - 71583232 n  - 70842624 n - 29869056
                *
                    4
                   x
               + 
                           8          7           6            5             4
                     3072 n  + 74240 n  + 780160 n  + 4655744 n  + 17254912 n
                   + 
                               3             2
                     40661888 n  + 59489152 n  + 49392384 n + 17814528
                *
                    2
                   x
               + 
                        9         8          7           6            5
                 - 256 n  - 6912 n  - 82368 n  - 568512 n  - 2504256 n
               + 
                          4             3             2
               - 7299648 n  - 14077952 n  - 17319168 n  - 12331008 n - 3870720
            *
               f(n + 2)
           + 
                           6           5           4            3            2
                     8192 n  + 131072 n  + 860160 n  + 2965504 n  + 5668864 n
                   + 
                     5701632 n + 2359296
                *
                    10
                   x
               + 
                              7           6            5             4
                     - 28672 n  - 540672 n  - 4321280 n  - 18980864 n
                   + 
                                 3             2
                     - 49496064 n  - 76644352 n  - 65273856 n - 23592960
                *
                    8
                   x
               + 
                            8           7            6             5
                     36864 n  + 800768 n  + 7542784 n  + 40245248 n
                   + 
                                4              3              2
                     133052416 n  + 279130112 n  + 362930176 n  + 267436032 n
                   + 
                     85524480
                *
                    6
                   x
               + 
                              9           8            7             6
                     - 21504 n  - 529408 n  - 5745664 n  - 36082688 n
                   + 
                                  5              4              3              2
                     - 144510976 n  - 382810112 n  - 670806016 n  - 749903872 n
                   + 
                     - 485376000 n - 138608640
                *
                    4
                   x
               + 
                           10           9            8             7
                     5632 n   + 155136 n  + 1906688 n  + 13770240 n
                   + 
                               6              5              4              3
                     64722432 n  + 206893056 n  + 455585792 n  + 682500096 n
                   + 
                                2
                     665804800 n  + 382009344 n + 97910784
                *
                    2
                   x
               + 
                        11          10           9            8            7
                 - 512 n   - 15616 n   - 214528 n  - 1752320 n  - 9457152 n
               + 
                             6             5              4              3
                 - 35414016 n  - 93908992 n  - 176377856 n  - 229990400 n
               + 
                              2
                 - 198344704 n  - 101842944 n - 23592960
            *
               f(n + 1)
           + 
                             10           9            8            7
                     - 8192 n   - 163840 n  - 1433600 n  - 7225344 n
                   + 
                                 6             5             4             3
                     - 23224320 n  - 49741824 n  - 71901184 n  - 69287936 n
                   + 
                                 2
                     - 42631168 n  - 15138816 n - 2359296
                *
                    6
                   x
               + 
                            11           10            9             8
                     16384 n   + 385024 n   + 4014080 n  + 24485888 n
                   + 
                               7              6              5              4
                     97026048 n  + 262053888 n  + 491995136 n  + 641884160 n
                   + 
                                3              2
                     570277888 n  + 328695808 n  + 110690304 n + 16515072
                *
                    4
                   x
               + 
                              12           11            10             9
                     - 10240 n   - 276480 n   - 3350528 n   - 24074240 n
                   + 
                                  8              7              6
                     - 114114560 n  - 375576576 n  - 879288320 n
                   + 
                                   5               4               3
                     - 1474308096 n  - 1756051456 n  - 1448587264 n
                   + 
                                  2
                     - 785539072 n  - 251510784 n - 35979264
                *
                    2
                   x
               + 
                       13          12           11            10             9
                 2048 n   + 62464 n   + 863232 n   + 7150592 n   + 39574528 n
               + 
                            8              7              6               5
                 154383360 n  + 436335616 n  + 903812096 n  + 1370926080 n
               + 
                             4               3              2
                 1502906368 n  + 1156861952 n  + 591962112 n  + 180486144 n
               + 
                 24772608
            *
               f(n)
         = 
           0
       ,
                           4              8        6        4        2
      f(0) = 1, f(1) = 16 x , f(2) = 256 x  - 512 x  + 384 x  - 128 x  + 16,
                   12          10          8          6          4
      f(3) = 4096 x   - 24576 x   + 55296 x  - 55296 x  + 20736 x ]
     ]
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:=4;
Type: PositiveInteger?
fricas
x2:=8/3;
Type: Fraction(Integer)
fricas
x3:=-16/35;
Type: Fraction(Integer)
fricas
x4:=128/693;
Type: Fraction(Integer)
fricas
x5:=-128/1287;
Type: Fraction(Integer)
fricas
x6:=14336/230945;
Type: Fraction(Integer)
fricas
x7:=-4096/96577;
Type: Fraction(Integer)
fricas
x8:=360448/11700675;
Type: Fraction(Integer)
fricas
x9:=-2342912/100180065;
Type: Fraction(Integer)
fricas
x10:=6815744/371255535;
Type: Fraction(Integer)
fricas
x11:=-17825792/1204952175;
Type: Fraction(Integer)
fricas
x12:=2709520384/222548928855;
Type: Fraction(Integer)
fricas
x13:=-18966642688/1860438674025;
Type: Fraction(Integer)
fricas
x14:=58653147136/6771996773451;
Type: Fraction(Integer)
fricas
x15:=-586531471360/78736520890181;
Type: Fraction(Integer)
fricas
x16:=148176371712/22884013460693;
Type: Fraction(Integer)
fricas
x17:=-1790464491520/315208830571481;
Type: Fraction(Integer)
fricas
x18:=2131368930705408/424299741297447197;
Type: Fraction(Integer)
fricas
x19:=-3397834527211520/759460881481985319;
Type: Fraction(Integer)
fricas
x20:=5436535243538432/1355691431937597989;
Type: Fraction(Integer)
fricas
l := [x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20];
Type: List(Fraction(Integer))
fricas
guessHolo(l)
\begin{equation*}
\label{eq19}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))
fricas
guess(l)
\begin{equation*}
\label{eq20}\left[ 4 \ {\left(\prod_{
\displaystyle
{{p_{20}}= 0}}^{
\displaystyle
{n - 1}}{\left[{{{f \left({p_{20}}\right)}\mbox{\rm :}}{{{{\left({{1
6}\ {{p_{20}}^{4}}}+{{32}\ {{p_{20}}^{3}}}+{{19}\ {{p_{20}}^{2}}}+{3 \ {p_{20}}}\right)}\ {f \left({p_{20}}\right)}}+{{16}\ {{p_{20}}^{4}}}-{4 \ {{p_{20}}^{2}}}}= 0}}\right]}\right)}\right]\end{equation*}
Type: List(Expression(Integer))
fricas
guess(l, [guessRat], [guessSum, guessProduct], maxLevel==8)
\begin{equation*}
\label{eq21}\left[ 4 \ {\left(\prod_{
\displaystyle
{{p_{20}}= 0}}^{
\displaystyle
{n - 1}}{\left[{{{f \left({p_{20}}\right)}\mbox{\rm :}}{{{{\left({{1
6}\ {{p_{20}}^{4}}}+{{32}\ {{p_{20}}^{3}}}+{{19}\ {{p_{20}}^{2}}}+{3 \ {p_{20}}}\right)}\ {f \left({p_{20}}\right)}}+{{16}\ {{p_{20}}^{4}}}-{4 \ {{p_{20}}^{2}}}}= 0}}\right]}\right)}\right]\end{equation*}
Type: List(Expression(Integer))
fricas
guessPRec(l)
\begin{equation*}
\label{eq22}\begin{array}{@{}l}
\displaystyle
\left[ \left[{{{f \left({n}\right)}\mbox{\rm :}}{{{{\left({{1
6}\ {{n}^{4}}}+{{32}\ {{n}^{3}}}+{{19}\ {{n}^{2}}}+{3 \  n}\right)}\ {f \left({n + 1}\right)}}+{{\left({{16}\ {{n}^{4}}}-{4 \ {{n}^{2}}}\right)}\ {f \left({n}\right)}}}= 0}}, \right.
\
\
\displaystyle
\left.\:{{f \left({0}\right)}= 4}\right] \right] 
\end{array}
\end{equation*}
Type: List(Expression(Integer))
fricas
guessADE(l)
\begin{equation*}
\label{eq23}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))
fricas
guessPRec(l)
\begin{equation*}
\label{eq24}\begin{array}{@{}l}
\displaystyle
\left[ \left[{{{f \left({n}\right)}\mbox{\rm :}}{{{{\left({{1
6}\ {{n}^{4}}}+{{32}\ {{n}^{3}}}+{{19}\ {{n}^{2}}}+{3 \  n}\right)}\ {f \left({n + 1}\right)}}+{{\left({{16}\ {{n}^{4}}}-{4 \ {{n}^{2}}}\right)}\ {f \left({n}\right)}}}= 0}}, \right.
\
\
\displaystyle
\left.\:{{f \left({0}\right)}= 4}\right] \right] 
\end{array}
\end{equation*}
Type: List(Expression(Integer))
fricas
guessHolo(l)
\begin{equation*}
\label{eq25}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))
fricas
guessAlg(l)
\begin{equation*}
\label{eq26}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))

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