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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 (- 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))

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))

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))

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:=32;
Type: PositiveInteger?
fricas
x2:=87;
Type: PositiveInteger?
fricas
x3:=192;
Type: PositiveInteger?
fricas
x4:=367;
Type: PositiveInteger?
fricas
x5:=640;
Type: PositiveInteger?
fricas
x6:=1039;
Type: PositiveInteger?
fricas
x7:=1600;
Type: PositiveInteger?
fricas
x8:=2359;
Type: PositiveInteger?
fricas
x9:=3360;
Type: PositiveInteger?
fricas
x10:=4647;
Type: PositiveInteger?
fricas
x11:=6272;
Type: PositiveInteger?
fricas
x12:=8287;
Type: PositiveInteger?
fricas
l := [x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12];
Type: List(PositiveInteger?)
fricas
guessHolo(l)
\begin{equation*} \label{eq19}\left[ \right]?\end{equation*}
Type: List(Expression(Integer))
fricas
guess(l)
\begin{equation*} \label{eq20}\left[{\sum_{ \displaystyle {{s_{12}}= 0}}^{ \displaystyle {n - 1}}{\left({\sum_{ \displaystyle {{s_{11}}= 0}}^{ \displaystyle {{s_{12}}- 1}}{\left({\sum_{ \displaystyle {{s_{10}}= 0}}^{ \displaystyle {{s_{11}}- 1}}{\left({\sum_{ \displaystyle {{s_{9}}= 0}}^{ \displaystyle {{s_{10}}- 1}}{\left({\sum_{ \displaystyle {{s_{8}}= 0}}^{ \displaystyle {{s_{9}}- 1}}{\left(-{8 \ {{\left(- 1 \right)}^{s_{8}}}}\right)}}+ 8 \right)}}+{20}\right)}}+{50}\right)}}+{55}\right)}}+{32}\right]\end{equation*}
Type: List(Expression(Integer))
fricas
guess(l, [guessRat], [guessSum, guessProduct], maxLevel==8)
\begin{equation*} \label{eq21}\left[{\sum_{ \displaystyle {{s_{12}}= 0}}^{ \displaystyle {n - 1}}{\left({\sum_{ \displaystyle {{s_{11}}= 0}}^{ \displaystyle {{s_{12}}- 1}}{\left({\sum_{ \displaystyle {{s_{10}}= 0}}^{ \displaystyle {{s_{11}}- 1}}{\left({\sum_{ \displaystyle {{s_{9}}= 0}}^{ \displaystyle {{s_{10}}- 1}}{\left({\sum_{ \displaystyle {{s_{8}}= 0}}^{ \displaystyle {{s_{9}}- 1}}{\left(-{8 \ {{\left(- 1 \right)}^{s_{8}}}}\right)}}+ 8 \right)}}+{20}\right)}}+{50}\right)}}+{55}\right)}}+{32}\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(n + 3 \right)}\ {f \left({n + 2}\right)}}-{4 \ {f \left({n + 1}\right)}}+{{\left(- n - 7 \right)}\ {f \left({n}\right)}}- 4}= 0}}, \right. \ \ \displaystyle \left.\:{{f \left({0}\right)}={32}}, \:{{f \left({1}\right)}={8 7}}\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(n + 3 \right)}\ {f \left({n + 2}\right)}}-{4 \ {f \left({n + 1}\right)}}+{{\left(- n - 7 \right)}\ {f \left({n}\right)}}- 4}= 0}}, \right. \ \ \displaystyle \left.\:{{f \left({0}\right)}={32}}, \:{{f \left({1}\right)}={8 7}}\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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Package amsmath Warning: Foreign command \over; (amsmath) \frac or \genfrac should be used instead (amsmath) on input line 189.

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] Missing \right. inserted. <inserted text> \right . l.291 \

Extra \right. l.295 7}}\right] \right]

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Extra \right. l.308 7}}\right] \right]

[28] [29] [30] (./8371586242334833778-16.0px.aux) ) (see the transcript file for additional information) Output written on 8371586242334833778-16.0px.dvi (30 pages, 12128 bytes). Transcript written on 8371586242334833778-16.0px.log.




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