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Edit detail for SandBox.GuessingSequence revision 349 of 349

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349
Editor: beccaria
Time: 2017/05/03 15:55:18 GMT+0
Note:

changed:
-x1:=6;
-x2:=35;
-x3:=138;
-x4:=379;
-x5:=898;
-x6:=1834;
-x7:=3460;
-x8:=6018;
-x9:=9958;
-x10:=15653;
-x11:=23774;
-x12:=34853;
-x13:=49826;
x1:=-2;
x2:=-1;
x3:=10;
x4:=79;
x5:=250;
x6:=658;
x7:=1412;
x8:=2778;
x9:=4958;
x10:=8393;
x11:=13406;
x12:=20657;
x13:=30618;

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)

\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 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]

\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}}{{{4 \ {p_{7}}}+ 2}\over{{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}}{{{{27}\ {{p_{7}}^{2}}}+{{54}\ {p_{7}}}+{24}}\over{{{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}\left[ -{1 \over 3}, \: -{{11}\over{25}}, \: -{{23}\over{49}}, \: -{{13}\over{27}}, \: -{{59}\over{121}}, \: -{{83}\over{16
9}}\right](8)
Type: List(Fraction(Integer))
fricas
guess(l, [guessRat], [guessSum, guessProduct], maxLevel==2)

\label{eq9}\left[{{-{2 \ {{n}^{2}}}-{6 \  n}- 3}\over{{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 (- 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 (- 88n - 1096n - 5016n - 9968n - 7232)x + 8n + 124n + 3 2 752n + 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 3392n + 56384n + 372096n + 1218048n + 1976896n + 1272064 * 4 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)

\label{eq15}\left[ \right](15)
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)

\label{eq16}\left[ \right](16)
Type: List(Expression(Integer))

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))

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:=-2;
Type: Integer
fricas
x2:=-1;
Type: Integer
fricas
x3:=10;
fricas
x4:=79;
fricas
x5:=250;
fricas
x6:=658;
fricas
x7:=1412;
fricas
x8:=2778;
fricas
x9:=4958;
fricas
x10:=8393;
fricas
x11:=13406;
fricas
x12:=20657;
fricas
x13:=30618;
fricas
l := [x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13];
Type: List(Integer)
fricas
guessHolo(l)

\label{eq19}\left[ \right](19)
Type: List(Expression(Integer))
fricas
guess(l)

\label{eq20}\left[{\sum_{
\displaystyle
{{s_{13}}= 0}}^{
\displaystyle
{n - 1}}{\left({\sum_{
\displaystyle
{{s_{12}}= 0}}^{
\displaystyle
{{s_{13}}- 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(-{{212}\ {\left(\prod_{
\displaystyle
{{p_{7}}= 0}}^{
\displaystyle
{{s_{8}}- 1}}{{-{2 \ {{p_{7}}^{2}}}-{{24}\ {p_{7}}}-{75}}\over{{2 \ {{p_{7}}^{2}}}+{{20}\ {p_{7}}}+{53}}}\right)}}\right)}}+{95}\right)}}- 4 \right)}}+{48}\right)}}+{10}\right)}}+ 1 \right)}}- 2 \right](20)
Type: List(Expression(Integer))
fricas
guess(l, [guessRat], [guessSum, guessProduct], maxLevel==4)

\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))