Av(1243, 2413, 41352, 531642)
Counting Sequence
1, 1, 2, 6, 22, 87, 353, 1447, 5971, 24795, 103626, 435831, 1844051, 7845963, 33553795, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x^{4}-6 x^{3}+12 x^{2}-8 x +2\right) \left(x -1\right)^{2} F \left(x
\right)^{4}+x \left(x -1\right) \left(3 x^{5}-16 x^{4}+26 x^{3}-12 x^{2}+1\right) F \left(x
\right)^{3}+\left(3 x^{7}-22 x^{6}+67 x^{5}-112 x^{4}+109 x^{3}-57 x^{2}+14 x -1\right) F \left(x
\right)^{2}+x \left(3 x^{2}-9 x +5\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(x -1\right)^{6} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 87\)
\(\displaystyle a \! \left(6\right) = 353\)
\(\displaystyle a \! \left(7\right) = 1447\)
\(\displaystyle a \! \left(8\right) = 5971\)
\(\displaystyle a \! \left(9\right) = 24795\)
\(\displaystyle a \! \left(10\right) = 103626\)
\(\displaystyle a \! \left(11\right) = 435831\)
\(\displaystyle a \! \left(12\right) = 1844051\)
\(\displaystyle a \! \left(13\right) = 7845963\)
\(\displaystyle a \! \left(14\right) = 33553795\)
\(\displaystyle a \! \left(15\right) = 144169233\)
\(\displaystyle a \! \left(16\right) = 622113535\)
\(\displaystyle a \! \left(17\right) = 2695141249\)
\(\displaystyle a \! \left(18\right) = 11718545059\)
\(\displaystyle a \! \left(19\right) = 51124178941\)
\(\displaystyle a \! \left(20\right) = 223734228330\)
\(\displaystyle a \! \left(21\right) = 981964657716\)
\(\displaystyle a \! \left(22\right) = 4321455087749\)
\(\displaystyle a \! \left(23\right) = 19065862627305\)
\(\displaystyle a \! \left(24\right) = 84314832161621\)
\(\displaystyle a \! \left(25\right) = 373686674642073\)
\(\displaystyle a \! \left(26\right) = 1659617316970834\)
\(\displaystyle a \! \left(27\right) = 7385000956733269\)
\(\displaystyle a \! \left(28\right) = 32921934614253250\)
\(\displaystyle a \! \left(29\right) = 147016549596884630\)
\(\displaystyle a \! \left(30\right) = 657583091600257626\)
\(\displaystyle a \! \left(31\right) = 2945772787729038122\)
\(\displaystyle a \! \left(32\right) = 13215257422562170770\)
\(\displaystyle a \! \left(33\right) = 59367123643322539835\)
\(\displaystyle a \! \left(34\right) = 267041665825159684625\)
\(\displaystyle a \! \left(35\right) = 1202667393947188117460\)
\(\displaystyle a \! \left(36\right) = 5422735213338557226711\)
\(\displaystyle a \! \left(37\right) = 24477797177070015216274\)
\(\displaystyle a \! \left(38\right) = 110607290359376284431921\)
\(\displaystyle a \! \left(39\right) = 500300160930586288188948\)
\(\displaystyle a \! \left(40\right) = 2265126280896088417185616\)
\(\displaystyle a \! \left(41\right) = 10264784292318376654965957\)
\(\displaystyle a \! \left(42\right) = 46556983792959056044246894\)
\(\displaystyle a \! \left(43\right) = 211339436856584118907069194\)
\(\displaystyle a \! \left(44\right) = 960110234004451264454551756\)
\(\displaystyle a \! \left(45\right) = 4365073962778444660854682932\)
\(\displaystyle a \! \left(46\right) = 19859946586465393722287609817\)
\(\displaystyle a \! \left(47\right) = 90420591395332384446730505959\)
\(\displaystyle a \! \left(48\right) = 411952378584666583926775152437\)
\(\displaystyle a \! \left(49\right) = 1878042484422960327036346699907\)
\(\displaystyle a \! \left(50\right) = 8567054355198621971890480699096\)
\(\displaystyle a \! \left(51\right) = 39103438515578915821809721652200\)
\(\displaystyle a \! \left(52\right) = 178585366355930743077672827404384\)
\(\displaystyle a \! \left(53\right) = 816046674275181730631551681919067\)
\(\displaystyle a \! \left(n +54\right) = \frac{\left(299 n +15713\right) a \! \left(n +53\right)}{4 n +220}-\frac{\left(5257 n^{2}+546535 n +14205126\right) a \! \left(n +52\right)}{2 \left(n +54\right) \left(n +55\right)}-\frac{\left(44235685155961175137 n^{3}+4286241045974050624578 n^{2}+138417351189770619616529 n +1489753874582570149435038\right) a \! \left(n +32\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(23288650489030052095 n^{3}+2324248672019727334206 n^{2}+77308189679437457554235 n +856984109372086729243416\right) a \! \left(n +33\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(11169683584308834676 n^{3}+1147206790461797102475 n^{2}+39268253555824497965921 n +447960763583266301441520\right) a \! \left(n +34\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(1623683299908221275 n^{3}+171481086289676412986 n^{2}+6035656913526794455417 n +70798736837128293131710\right) a \! \left(n +35\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(963555294481623337 n^{3}+104563421858393757885 n^{2}+3781558889271530481269 n +45577328503601715772899\right) a \! \left(n +36\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(229953362330413077 n^{3}+25622877999683997116 n^{2}+951478240678027904199 n +11774740957298859846264\right) a \! \left(n +37\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(74264426761402485 n^{3}+8491295357915145440 n^{2}+323552962281704958605 n +4108598017367998552438\right) a \! \left(n +38\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(32349724146038926 n^{3}+3793267709791417461 n^{2}+148228346202442159478 n +1930288516187455754271\right) a \! \left(n +39\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(2805772201076593 n^{3}+337223481390591296 n^{2}+13506914541173188045 n +180287518105797439910\right) a \! \left(n +40\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(3910081091629051 n^{3}+481480754696020038 n^{2}+19758108821544767561 n +270198715285399573974\right) a \! \left(n +41\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(67307869977133 n^{3}+8488505971619997 n^{2}+356757707534111157 n +4996782712807793105\right) a \! \left(n +42\right)}{8 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(12325422308298 n^{3}+1591588486378765 n^{2}+68492640868485408 n +982290039136649245\right) a \! \left(n +43\right)}{8 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(2994663835645 n^{3}+395898728513376 n^{2}+17442810159804326 n +256120361437357041\right) a \! \left(n +44\right)}{4 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(1715420844451 n^{3}+232168633561518 n^{2}+10472478182963093 n +157437414976983234\right) a \! \left(n +45\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(108761973433 n^{3}+15069884614284 n^{2}+695945530222445 n +10712061329301774\right) a \! \left(n +46\right)}{8 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(2040162804 n^{3}+289360368695 n^{2}+13679304782390 n +215546050985655\right) a \! \left(n +47\right)}{4 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(305398838 n^{3}+44315899287 n^{2}+2143460964037 n +34556982803358\right) a \! \left(n +48\right)}{2 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(106829729 n^{3}+15843554490 n^{2}+783219255955 n +12905792847330\right) a \! \left(n +49\right)}{8 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(1985288 n^{3}+300489834 n^{2}+15160291183 n +254951229399\right) a \! \left(n +50\right)}{2 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(118799 n^{3}+18323451 n^{2}+942043351 n +16143732471\right) a \! \left(n +51\right)}{2 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(23203883 n^{3}+202484928 n^{2}+588455119 n +568166910\right) a \! \left(n +2\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(1324937572 n^{3}+15463284279 n^{2}+60238448231 n +78216068430\right) a \! \left(n +3\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(6250596270 n^{3}+91586737201 n^{2}+448090218863 n +731361920874\right) a \! \left(n +4\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(30609458247009336 n^{3}+1179474471196475195 n^{2}+15151192493183126961 n +64881421932352872044\right) a \! \left(n +12\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{45 \left(2 n +3\right) \left(6733 n^{2}+29174 n +31745\right) a \! \left(n +1\right)}{4 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(175881031057546222627 n^{3}+15507272144766283538394 n^{2}+455697564046538503304663 n +4463170035667156803848406\right) a \! \left(n +29\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(47200841046185098853 n^{3}+3886731601107341615681 n^{2}+106672460787446474888326 n +975784465326766158720499\right) a \! \left(n +27\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(233208776068338660127 n^{3}+19882787689245840642393 n^{2}+564987418427153221920116 n +5350940285394945895955118\right) a \! \left(n +28\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(315071456897315755981 n^{3}+25026290392857512744622 n^{2}+662552484953353047502913 n +5846305648080631466378028\right) a \! \left(n +26\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(321133674564010395859 n^{3}+24571738712191282768497 n^{2}+626650492261529570584736 n +5326666690256543055900786\right) a \! \left(n +25\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(299808765606580917355 n^{3}+22065929402054315895303 n^{2}+541305338830982380747298 n +4425943976606642723179548\right) a \! \left(n +24\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(256267731295763700950 n^{3}+18114010690732069202529 n^{2}+426756315247947826485697 n +3351130946258492568754284\right) a \! \left(n +23\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(200419363462759988680 n^{3}+13581921029947930580685 n^{2}+306781823333772055462655 n +2309651924994689992329042\right) a \! \left(n +22\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(143278244089151702828 n^{3}+9291712442733592456719 n^{2}+200844689755634497290493 n +1447023935497251109682070\right) a \! \left(n +21\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(93520040775090004487 n^{3}+5792057310522084794370 n^{2}+119567281731331387626211 n +822706540059653625218544\right) a \! \left(n +20\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(55652891570641606763 n^{3}+3284437500949912518498 n^{2}+64608277647596395237603 n +423614631915943390881564\right) a \! \left(n +19\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(30142788557254728176 n^{3}+1690961374957876123995 n^{2}+31618478613140792355547 n +197063810617776569926398\right) a \! \left(n +18\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(14829162396779514683 n^{3}+788601262542672623982 n^{2}+13978463466306045738985 n +82589303914440001603242\right) a \! \left(n +17\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(2203691076241577861 n^{3}+110752963594804705264 n^{2}+1855353809921615708805 n +10360137398153020419904\right) a \! \left(n +16\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(2663665586922502229 n^{3}+126082020308139550056 n^{2}+1989304526382609106111 n +10462213239504012945888\right) a \! \left(n +15\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(966935019386040212 n^{3}+42937921271882592405 n^{2}+635580673468158514207 n +3136057078520578900446\right) a \! \left(n +14\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(52521158789046874 n^{3}+2178214482752605915 n^{2}+30114046809731598135 n +138782399918323398726\right) a \! \left(n +13\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(23813975655058906 n^{3}+847400472583644105 n^{2}+10053192868340213411 n +39761282732580601410\right) a \! \left(n +11\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(1822032705140176 n^{3}+59443321480904427 n^{2}+646629169119787005 n +2345222893178734050\right) a \! \left(n +10\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(1103352622216432 n^{3}+32717841148928685 n^{2}+323537942752741217 n +1066817992655540952\right) a \! \left(n +9\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(64781683137051 n^{3}+1727628694915868 n^{2}+15367616254679701 n +45587864526331896\right) a \! \left(n +8\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(14792824561088 n^{3}+350166843351279 n^{2}+2765476270430233 n +7284948622036656\right) a \! \left(n +7\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(961176520459 n^{3}+19862557320363 n^{2}+136985056966376 n +315174453370476\right) a \! \left(n +6\right)}{8 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(209713891603 n^{3}+3702519650433 n^{2}+21822813126368 n +42915570066144\right) a \! \left(n +5\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(76670900390438371655 n^{3}+7206155093606574993615 n^{2}+225731410990225347544174 n +2356654329853091249558232\right) a \! \left(n +31\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(60710555628696931154 n^{3}+5529477356823724603923 n^{2}+167851123960823807655163 n +1698185250358751335682811\right) a \! \left(n +30\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{2625 \left(2 n +3\right) \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{4 \left(n +55\right) \left(n +54\right) \left(n +53\right)}, \quad n \geq 54\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 87\)
\(\displaystyle a \! \left(6\right) = 353\)
\(\displaystyle a \! \left(7\right) = 1447\)
\(\displaystyle a \! \left(8\right) = 5971\)
\(\displaystyle a \! \left(9\right) = 24795\)
\(\displaystyle a \! \left(10\right) = 103626\)
\(\displaystyle a \! \left(11\right) = 435831\)
\(\displaystyle a \! \left(12\right) = 1844051\)
\(\displaystyle a \! \left(13\right) = 7845963\)
\(\displaystyle a \! \left(14\right) = 33553795\)
\(\displaystyle a \! \left(15\right) = 144169233\)
\(\displaystyle a \! \left(16\right) = 622113535\)
\(\displaystyle a \! \left(17\right) = 2695141249\)
\(\displaystyle a \! \left(18\right) = 11718545059\)
\(\displaystyle a \! \left(19\right) = 51124178941\)
\(\displaystyle a \! \left(20\right) = 223734228330\)
\(\displaystyle a \! \left(21\right) = 981964657716\)
\(\displaystyle a \! \left(22\right) = 4321455087749\)
\(\displaystyle a \! \left(23\right) = 19065862627305\)
\(\displaystyle a \! \left(24\right) = 84314832161621\)
\(\displaystyle a \! \left(25\right) = 373686674642073\)
\(\displaystyle a \! \left(26\right) = 1659617316970834\)
\(\displaystyle a \! \left(27\right) = 7385000956733269\)
\(\displaystyle a \! \left(28\right) = 32921934614253250\)
\(\displaystyle a \! \left(29\right) = 147016549596884630\)
\(\displaystyle a \! \left(30\right) = 657583091600257626\)
\(\displaystyle a \! \left(31\right) = 2945772787729038122\)
\(\displaystyle a \! \left(32\right) = 13215257422562170770\)
\(\displaystyle a \! \left(33\right) = 59367123643322539835\)
\(\displaystyle a \! \left(34\right) = 267041665825159684625\)
\(\displaystyle a \! \left(35\right) = 1202667393947188117460\)
\(\displaystyle a \! \left(36\right) = 5422735213338557226711\)
\(\displaystyle a \! \left(37\right) = 24477797177070015216274\)
\(\displaystyle a \! \left(38\right) = 110607290359376284431921\)
\(\displaystyle a \! \left(39\right) = 500300160930586288188948\)
\(\displaystyle a \! \left(40\right) = 2265126280896088417185616\)
\(\displaystyle a \! \left(41\right) = 10264784292318376654965957\)
\(\displaystyle a \! \left(42\right) = 46556983792959056044246894\)
\(\displaystyle a \! \left(43\right) = 211339436856584118907069194\)
\(\displaystyle a \! \left(44\right) = 960110234004451264454551756\)
\(\displaystyle a \! \left(45\right) = 4365073962778444660854682932\)
\(\displaystyle a \! \left(46\right) = 19859946586465393722287609817\)
\(\displaystyle a \! \left(47\right) = 90420591395332384446730505959\)
\(\displaystyle a \! \left(48\right) = 411952378584666583926775152437\)
\(\displaystyle a \! \left(49\right) = 1878042484422960327036346699907\)
\(\displaystyle a \! \left(50\right) = 8567054355198621971890480699096\)
\(\displaystyle a \! \left(51\right) = 39103438515578915821809721652200\)
\(\displaystyle a \! \left(52\right) = 178585366355930743077672827404384\)
\(\displaystyle a \! \left(53\right) = 816046674275181730631551681919067\)
\(\displaystyle a \! \left(n +54\right) = \frac{\left(299 n +15713\right) a \! \left(n +53\right)}{4 n +220}-\frac{\left(5257 n^{2}+546535 n +14205126\right) a \! \left(n +52\right)}{2 \left(n +54\right) \left(n +55\right)}-\frac{\left(44235685155961175137 n^{3}+4286241045974050624578 n^{2}+138417351189770619616529 n +1489753874582570149435038\right) a \! \left(n +32\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(23288650489030052095 n^{3}+2324248672019727334206 n^{2}+77308189679437457554235 n +856984109372086729243416\right) a \! \left(n +33\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(11169683584308834676 n^{3}+1147206790461797102475 n^{2}+39268253555824497965921 n +447960763583266301441520\right) a \! \left(n +34\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(1623683299908221275 n^{3}+171481086289676412986 n^{2}+6035656913526794455417 n +70798736837128293131710\right) a \! \left(n +35\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(963555294481623337 n^{3}+104563421858393757885 n^{2}+3781558889271530481269 n +45577328503601715772899\right) a \! \left(n +36\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(229953362330413077 n^{3}+25622877999683997116 n^{2}+951478240678027904199 n +11774740957298859846264\right) a \! \left(n +37\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(74264426761402485 n^{3}+8491295357915145440 n^{2}+323552962281704958605 n +4108598017367998552438\right) a \! \left(n +38\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(32349724146038926 n^{3}+3793267709791417461 n^{2}+148228346202442159478 n +1930288516187455754271\right) a \! \left(n +39\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(2805772201076593 n^{3}+337223481390591296 n^{2}+13506914541173188045 n +180287518105797439910\right) a \! \left(n +40\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(3910081091629051 n^{3}+481480754696020038 n^{2}+19758108821544767561 n +270198715285399573974\right) a \! \left(n +41\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(67307869977133 n^{3}+8488505971619997 n^{2}+356757707534111157 n +4996782712807793105\right) a \! \left(n +42\right)}{8 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(12325422308298 n^{3}+1591588486378765 n^{2}+68492640868485408 n +982290039136649245\right) a \! \left(n +43\right)}{8 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(2994663835645 n^{3}+395898728513376 n^{2}+17442810159804326 n +256120361437357041\right) a \! \left(n +44\right)}{4 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(1715420844451 n^{3}+232168633561518 n^{2}+10472478182963093 n +157437414976983234\right) a \! \left(n +45\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(108761973433 n^{3}+15069884614284 n^{2}+695945530222445 n +10712061329301774\right) a \! \left(n +46\right)}{8 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(2040162804 n^{3}+289360368695 n^{2}+13679304782390 n +215546050985655\right) a \! \left(n +47\right)}{4 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(305398838 n^{3}+44315899287 n^{2}+2143460964037 n +34556982803358\right) a \! \left(n +48\right)}{2 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(106829729 n^{3}+15843554490 n^{2}+783219255955 n +12905792847330\right) a \! \left(n +49\right)}{8 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(1985288 n^{3}+300489834 n^{2}+15160291183 n +254951229399\right) a \! \left(n +50\right)}{2 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(118799 n^{3}+18323451 n^{2}+942043351 n +16143732471\right) a \! \left(n +51\right)}{2 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(23203883 n^{3}+202484928 n^{2}+588455119 n +568166910\right) a \! \left(n +2\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(1324937572 n^{3}+15463284279 n^{2}+60238448231 n +78216068430\right) a \! \left(n +3\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(6250596270 n^{3}+91586737201 n^{2}+448090218863 n +731361920874\right) a \! \left(n +4\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(30609458247009336 n^{3}+1179474471196475195 n^{2}+15151192493183126961 n +64881421932352872044\right) a \! \left(n +12\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{45 \left(2 n +3\right) \left(6733 n^{2}+29174 n +31745\right) a \! \left(n +1\right)}{4 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(175881031057546222627 n^{3}+15507272144766283538394 n^{2}+455697564046538503304663 n +4463170035667156803848406\right) a \! \left(n +29\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(47200841046185098853 n^{3}+3886731601107341615681 n^{2}+106672460787446474888326 n +975784465326766158720499\right) a \! \left(n +27\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(233208776068338660127 n^{3}+19882787689245840642393 n^{2}+564987418427153221920116 n +5350940285394945895955118\right) a \! \left(n +28\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(315071456897315755981 n^{3}+25026290392857512744622 n^{2}+662552484953353047502913 n +5846305648080631466378028\right) a \! \left(n +26\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(321133674564010395859 n^{3}+24571738712191282768497 n^{2}+626650492261529570584736 n +5326666690256543055900786\right) a \! \left(n +25\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(299808765606580917355 n^{3}+22065929402054315895303 n^{2}+541305338830982380747298 n +4425943976606642723179548\right) a \! \left(n +24\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(256267731295763700950 n^{3}+18114010690732069202529 n^{2}+426756315247947826485697 n +3351130946258492568754284\right) a \! \left(n +23\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(200419363462759988680 n^{3}+13581921029947930580685 n^{2}+306781823333772055462655 n +2309651924994689992329042\right) a \! \left(n +22\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(143278244089151702828 n^{3}+9291712442733592456719 n^{2}+200844689755634497290493 n +1447023935497251109682070\right) a \! \left(n +21\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(93520040775090004487 n^{3}+5792057310522084794370 n^{2}+119567281731331387626211 n +822706540059653625218544\right) a \! \left(n +20\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(55652891570641606763 n^{3}+3284437500949912518498 n^{2}+64608277647596395237603 n +423614631915943390881564\right) a \! \left(n +19\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(30142788557254728176 n^{3}+1690961374957876123995 n^{2}+31618478613140792355547 n +197063810617776569926398\right) a \! \left(n +18\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(14829162396779514683 n^{3}+788601262542672623982 n^{2}+13978463466306045738985 n +82589303914440001603242\right) a \! \left(n +17\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(2203691076241577861 n^{3}+110752963594804705264 n^{2}+1855353809921615708805 n +10360137398153020419904\right) a \! \left(n +16\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(2663665586922502229 n^{3}+126082020308139550056 n^{2}+1989304526382609106111 n +10462213239504012945888\right) a \! \left(n +15\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(966935019386040212 n^{3}+42937921271882592405 n^{2}+635580673468158514207 n +3136057078520578900446\right) a \! \left(n +14\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{3 \left(52521158789046874 n^{3}+2178214482752605915 n^{2}+30114046809731598135 n +138782399918323398726\right) a \! \left(n +13\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(23813975655058906 n^{3}+847400472583644105 n^{2}+10053192868340213411 n +39761282732580601410\right) a \! \left(n +11\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(1822032705140176 n^{3}+59443321480904427 n^{2}+646629169119787005 n +2345222893178734050\right) a \! \left(n +10\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(1103352622216432 n^{3}+32717841148928685 n^{2}+323537942752741217 n +1066817992655540952\right) a \! \left(n +9\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{3 \left(64781683137051 n^{3}+1727628694915868 n^{2}+15367616254679701 n +45587864526331896\right) a \! \left(n +8\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(14792824561088 n^{3}+350166843351279 n^{2}+2765476270430233 n +7284948622036656\right) a \! \left(n +7\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(961176520459 n^{3}+19862557320363 n^{2}+136985056966376 n +315174453370476\right) a \! \left(n +6\right)}{8 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(209713891603 n^{3}+3702519650433 n^{2}+21822813126368 n +42915570066144\right) a \! \left(n +5\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}+\frac{\left(76670900390438371655 n^{3}+7206155093606574993615 n^{2}+225731410990225347544174 n +2356654329853091249558232\right) a \! \left(n +31\right)}{32 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{\left(60710555628696931154 n^{3}+5529477356823724603923 n^{2}+167851123960823807655163 n +1698185250358751335682811\right) a \! \left(n +30\right)}{16 \left(n +55\right) \left(n +54\right) \left(n +53\right)}-\frac{2625 \left(2 n +3\right) \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{4 \left(n +55\right) \left(n +54\right) \left(n +53\right)}, \quad n \geq 54\)
This specification was found using the strategy pack "Insertion Row And Col Placements Req Corrob Expand Verified" and has 41 rules.
Found on January 22, 2022.Finding the specification took 10 seconds.
Copy 41 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{2}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{29}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{19}\! \left(x \right) F_{29}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{2}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{2}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right) F_{5}\! \left(x \right)\\
\end{align*}\)