MathAction SandBox.GuessingSequence

Topics

FrontPage
- SandBox
  - SandBox.GuessingSequence <-- You are here.

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)

$\label{eq1}\left[ \right]$

(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...

fricas

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

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)]

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[ 1, \: 1, \:{q + 1}, \:{{{q}^{2}}+ q + 1}, \:{{{q}^{4}}+{{q}^{3}}+{{q}^{2}}+ q + 1}, \: \right. \ \ \displaystyle \left.{{{q}^{6}}+{{q}^{5}}+{2 \ {{q}^{4}}}+{{q}^{3}}+{{q}^{2}}+ q + 1}, \: \right. \ \ \displaystyle \left.{{{q}^{9}}+{{q}^{8}}+{{q}^{7}}+{2 \ {{q}^{6}}}+{2 \ {{q}^{5}}}+{2 \ {{q}^{4}}}+{{q}^{3}}+{{q}^{2}}+ q + 1}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle {{q}^{12}}+{{q}^{11}}+{2 \ {{q}^{10}}}+{2 \ {{q}^{9}}}+{2 \ {{q}^{8}}}+{2 \ {{q}^{7}}}+{3 \ {{q}^{6}}}+{2 \ {{q}^{5}}}+{2 \ {{q}^{4}}}+ \ \ \displaystyle {{q}^{3}}+{{q}^{2}}+ q + 1$

(3)

Type: List(Polynomial(Integer))

fricas

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:

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])

$\label{eq5}\left[{\prod_{ \displaystyle {{p_{7}}= 0}}^{ \displaystyle {n - 1}}{\frac{{4 \ {p_{7}}}+ 2}{{p_{7}}+ 2}}}\right]$

(5)

Type: List(Expression(Integer))

fricas

guess(listB, [guessRat], [guessSum, guessProduct])

$\label{eq6}\left[ \right]$

(6)

Type: List(Expression(Integer))

fricas

guess(listC, [guessRat], [guessProduct]).1

$\label{eq7}\prod_{ \displaystyle {{p_{8}}= 0}}^{ \displaystyle {n - 1}}{\prod_{ \displaystyle {{p_{7}}= 0}}^{ \displaystyle {{p_{8}}- 1}}{\frac{{{27}\ {{p_{7}}^{2}}}+{{54}\ {p_{7}}}+{24}}{{{1 6}\ {{p_{7}}^{2}}}+{{32}\ {p_{7}}}+{12}}}}$

(7)

Type: Expression(Integer)

fricas

l := [-1/3,-11/25,-23/49,-13/27,-59/121,-83/169]

$\label{eq8}\begin{array}{@{}l} \displaystyle \left[ -{\frac{1}{3}}, \: -{\frac{11}{25}}, \: -{\frac{23}{49}}, \: -{\frac{13}{27}}, \: -{\frac{59}{121}}, \right. \ \ \displaystyle \left.\: -{\frac{83}{169}}\right]$

(8)

Type: List(Fraction(Integer))

fricas

guess(l, [guessRat], [guessSum, guessProduct], maxLevel==2)

$\label{eq9}\left[{\frac{-{2 \ {{n}^{2}}}-{6 \ n}- 3}{{4 \ {{n}^{2}}}+{{1 2}\ n}+ 9}}\right]$

(9)

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])

$\label{eq10}\left[ \right]$

(10)

Type: List(Expression(Integer))

fricas

guess(listE, [guessRat], [guessProduct])

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

(11)

Type: List(Expression(Integer))

fricas

li :=  [-86, -975, -100, -1728, -31213];

Type: List(Integer)

fricas

guess(li, [guessRat], [guessSum, guessProduct])

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

(12)

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]

$\label{eq13}\begin{array}{@{}l} \displaystyle \left[ 1, \: 1, \: 6, \:{54}, \:{660}, \:{10260}, \:{194040}, \:{4326840}, \:{111177360}, \: \right. \ \ \displaystyle \left.{3234848400}, \:{105135861600}, \:{3775206204000}\right]$

(13)

Type: List(PositiveInteger?)

fricas

guessPRec(l).1

$\label{eq14}\begin{array}{@{}l} \displaystyle \left[{{{f \left({n}\right)}\mbox{\rm :}}{{{f \left({n + 2}\right)}+{{\left(-{4 \ n}- 6 \right)}\ {f \left({n + 1}\right)}}+{{\left({2 \ {{n}^{2}}}+{4 \ n}\right)}\ {f \left({n}\right)}}}= 0}}, \right. \ \ \displaystyle \left.\:{{f \left({0}\right)}= 1}, \:{{f \left({1}\right)}= 1}\right]$

(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):

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)

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

(15)

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)

$\label{eq16}\left[ \right]$

(16)

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)

$\label{eq17}\left[ \right]$

(17)

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)

$\label{eq18}\left[ \right]$

(18)

Type: List(Expression(Integer))

fricas

x1:=1;

Type: PositiveInteger?

fricas

x2:=8;

Type: PositiveInteger?

fricas

x3:=11;

Type: PositiveInteger?

fricas

x4:=16;

Type: PositiveInteger?

fricas

x5:=12;

Type: PositiveInteger?

fricas

x6:=16;

Type: PositiveInteger?

fricas

x7:=12;

Type: PositiveInteger?

fricas

l := [x1,x2,x3,x4,x5,x6,x7];

Type: List(PositiveInteger?)

fricas

guessHolo(l)

$\label{eq19}\left[ \right]$

(19)

Type: List(Expression(Integer))

fricas

guess(l)

$\label{eq20}\left[ \right]$

(20)

Type: List(Expression(Integer))

fricas

guess(l, [guessRat], [guessSum, guessProduct], maxLevel==8)

$\label{eq21}\left[ \right]$

(21)

Type: List(Expression(Integer))

fricas

guessPRec(l)

$\label{eq22}\left[ \right]$

(22)

Type: List(Expression(Integer))

fricas

guessADE(l)

$\label{eq23}\left[ \right]$

(23)

Type: List(Expression(Integer))

fricas

guessPRec(l)

$\label{eq24}\left[ \right]$

(24)

Type: List(Expression(Integer))

fricas

guessHolo(l)

$\label{eq25}\left[ \right]$

(25)

Type: List(Expression(Integer))

fricas

guessAlg(l)

$\label{eq26}\left[ \right]$

(26)

Type: List(Expression(Integer))

Subject: (replying) Be Bold !!

	( 14 subscribers )