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 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. fricasguess([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... fricasguessExpRat [(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 fricasl := [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)) fricasguessPRec(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: fricaslistA := [1,1,2,5,14,42,132]; Type: List(PositiveInteger?) fricaslistB := [1,2,6,21,80, 322]; Type: List(PositiveInteger?) fricaslistC := [1,1,2,7,42,429,7436,218348]; Type: List(PositiveInteger?) fricasguess(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)) fricasguess(listB, [guessRat], [guessSum, guessProduct]) \begin{equation*} \label{eq6}\left[ \right]?\end{equation*} Type: List(Expression(Integer)) fricasguess(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) fricasl := [-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)) fricasguess(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)) fricaslistD := [1,1,2,6,26,162,1450,18626]; Type: List(PositiveInteger?) fricaslistE := [1,1,2,6,28,202,2252]; Type: List(PositiveInteger?) fricasguess(listD, [guessRat], [guessProduct]) \begin{equation*} \label{eq10}\left[ \right]?\end{equation*} Type: List(Expression(Integer)) fricasguess(listE, [guessRat], [guessProduct]) \begin{equation*} \label{eq11}\left[ \right]?\end{equation*} Type: List(Expression(Integer)) fricasli := [-86, -975, -100, -1728, -31213]; Type: List(Integer) fricasguess(li, [guessRat], [guessSum, guessProduct]) \begin{equation*} \label{eq12}\left[ \right]?\end{equation*} Type: List(Expression(Integer)) "Most" sequences arising in combinatorics are P-recursive: fricasl := [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?) fricasguessPRec(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) fricasguessPRec(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: fricasl := [hermiteH(n, x)^4 for n in 0..110]; Type: List(Polynomial(Integer)) fricasguessPRec(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 fricasguess([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 replyfricasguess([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 replyfricasguess([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 replyfricasguess([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)) fricasx1:=32; Type: PositiveInteger? fricasx2:=87; Type: PositiveInteger? fricasx3:=192; Type: PositiveInteger? fricasx4:=367; Type: PositiveInteger? fricasx5:=640; Type: PositiveInteger? fricasx6:=1039; Type: PositiveInteger? fricasx7:=1600; Type: PositiveInteger? fricasx8:=2359; Type: PositiveInteger? fricasx9:=3360; Type: PositiveInteger? fricasx10:=4647; Type: PositiveInteger? fricasx11:=6272; Type: PositiveInteger? fricasx12:=8287; Type: PositiveInteger? fricasl := [x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12]; Type: List(PositiveInteger?) fricasguessHolo(l) \begin{equation*} \label{eq19}\left[ \right]?\end{equation*} Type: List(Expression(Integer)) fricasguess(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)) fricasguess(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)) fricasguessPRec(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)) fricasguessADE(l) \begin{equation*} \label{eq23}\left[ \right]?\end{equation*} Type: List(Expression(Integer)) fricasguessPRec(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)) fricasguessHolo(l) \begin{equation*} \label{eq25}\left[ \right]?\end{equation*} Type: List(Expression(Integer)) fricasguessAlg(l) \begin{equation*} \label{eq26}\left[ \right]?\end{equation*} Type: List(Expression(Integer))  Some or all expressions may not have rendered properly, because Latex returned the following error:This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) \write18 enabled. %&-line parsing enabled. entering extended mode (./8371586242334833778-16.0px.tex LaTeX2e <2005/12/01> Babel and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, arabic, farsi, croatian, ukrainian, russian, bulgarian, czech, slov ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp eranto, uppersorbian, indonesian, polish, portuguese, spanish, catalan, galicia n, swedish, ukenglish, pinyin, loaded. 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