Av(13452, 13542, 14352, 14523, 14532, 15342, 15423, 15432, 24513, 25413)
Counting Sequence
1, 1, 2, 6, 24, 110, 544, 2814, 15014, 81982, 455894, 2573022, 14701354, 84871918, 494316702, ...
Implicit Equation for the Generating Function
\(\displaystyle 2 x^{3} \left(x -1\right) F \left(x
\right)^{5}-2 x^{2} \left(x^{2}-3 x +1\right) F \left(x
\right)^{4}-\left(x -1\right) \left(2 x^{2}+1\right) F \left(x
\right)^{3}+\left(-x^{2}+4 x -5\right) F \left(x
\right)^{2}+\left(-4 x +8\right) F \! \left(x \right)-4 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 110\)
\(\displaystyle a \! \left(6\right) = 544\)
\(\displaystyle a \! \left(7\right) = 2814\)
\(\displaystyle a \! \left(8\right) = 15014\)
\(\displaystyle a \! \left(9\right) = 81982\)
\(\displaystyle a \! \left(10\right) = 455894\)
\(\displaystyle a \! \left(11\right) = 2573022\)
\(\displaystyle a \! \left(12\right) = 14701354\)
\(\displaystyle a \! \left(13\right) = 84871918\)
\(\displaystyle a \! \left(14\right) = 494316702\)
\(\displaystyle a \! \left(15\right) = 2901059978\)
\(\displaystyle a \! \left(16\right) = 17139336394\)
\(\displaystyle a \! \left(17\right) = 101851923374\)
\(\displaystyle a \! \left(18\right) = 608406199902\)
\(\displaystyle a \! \left(19\right) = 3651110194354\)
\(\displaystyle a \! \left(20\right) = 22001788156210\)
\(\displaystyle a \! \left(21\right) = 133081427189222\)
\(\displaystyle a \! \left(22\right) = 807704347570678\)
\(\displaystyle a \! \left(23\right) = 4917360106158202\)
\(\displaystyle a \! \left(24\right) = 30022165036930698\)
\(\displaystyle a \! \left(25\right) = 183773180635729838\)
\(\displaystyle a \! \left(26\right) = 1127622885663073662\)
\(\displaystyle a \! \left(27\right) = 6934397783301056770\)
\(\displaystyle a \! \left(28\right) = 42731366510247561954\)
\(\displaystyle a \! \left(29\right) = 263824647059599069574\)
\(\displaystyle a \! \left(30\right) = 1631767403417108951814\)
\(\displaystyle a \! \left(31\right) = 10109384568340727836234\)
\(\displaystyle a \! \left(32\right) = 62729071246726359088090\)
\(\displaystyle a \! \left(33\right) = 389806433720715603160158\)
\(\displaystyle a \! \left(34\right) = 2425644833897932329065838\)
\(\displaystyle a \! \left(35\right) = 15113631742148392087327378\)
\(\displaystyle a \! \left(36\right) = 94284902341556014933816850\)
\(\displaystyle a \! \left(37\right) = 588868133361715365125620214\)
\(\displaystyle a \! \left(38\right) = 3681880926072536273923978838\)
\(\displaystyle a \! \left(39\right) = 23044777003175577310852255450\)
\(\displaystyle a \! \left(40\right) = 144378836234286920372919379466\)
\(\displaystyle a \! \left(41\right) = 905402482249504431366604747854\)
\(\displaystyle a \! \left(42\right) = 5682864878739801813357413075134\)
\(\displaystyle a \! \left(43\right) = 35699510062198123315151142313378\)
\(\displaystyle a \! \left(44\right) = 224444768335622620603114016499522\)
\(\displaystyle a \! \left(45\right) = 1412190201972625298892619316422118\)
\(\displaystyle a \! \left(46\right) = 8891982356014798595843457306523942\)
\(\displaystyle a \! \left(47\right) = 56028868212814763851004223148289130\)
\(\displaystyle a \! \left(48\right) = 353280777686400836323157874709894202\)
\(\displaystyle a \! \left(49\right) = 2229004312186824174560207630375785534\)
\(\displaystyle a \! \left(50\right) = 14072560730489362668118100304584882446\)
\(\displaystyle a \! \left(51\right) = 88898807148911691278014610490609440690\)
\(\displaystyle a \! \left(52\right) = 561913152555701557453262540645872882290\)
\(\displaystyle a \! \left(n +53\right) = -\frac{684571623424 n \left(n +3\right) \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{671744 \left(31367421 n +123968152\right) \left(n +3\right) \left(n +2\right) \left(n +1\right) a \! \left(n +1\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(142325 n^{3}+21853936 n^{2}+1118300001 n +19070710440\right) a \! \left(n +51\right)}{125 \left(n +54\right) \left(n +53\right) \left(2 n +109\right)}+\frac{\left(1929 n^{2}+201704 n +5269890\right) a \! \left(n +52\right)}{25 \left(2 n +109\right) \left(n +54\right)}-\frac{256 \left(61750953173387 n^{4}+1301222274794262 n^{3}+10203378066585886 n^{2}+35242192236622737 n +45184497177052260\right) a \! \left(n +4\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{128 \left(616254820865543 n^{4}+15407217124459762 n^{3}+143794240050268804 n^{2}+593186574910597037 n +911814806572939032\right) a \! \left(n +5\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{64 \left(5163068112875357 n^{4}+150241148039818720 n^{3}+1634273609803891972 n^{2}+7869990955134689459 n +14146758080334252378\right) a \! \left(n +6\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{32 \left(37365992537243693 n^{4}+1242830512620975881 n^{3}+15460069981490023243 n^{2}+85194986825310888553 n +175395080679324695166\right) a \! \left(n +7\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{32 \left(117402168039830971 n^{4}+4386231178759590298 n^{3}+61300470875537879831 n^{2}+379673530848347682362 n +879020130259163162988\right) a \! \left(n +8\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{16 \left(635004810060654751 n^{4}+26237347800960043901 n^{3}+405644950057376848463 n^{2}+2780569185206532680563 n +7128568563302630455290\right) a \! \left(n +9\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{16 \left(1457584839532261818 n^{4}+65741172673748813335 n^{3}+1109971170358619567397 n^{2}+8313333085661486372954 n +23301336196500640442304\right) a \! \left(n +10\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{16 \left(2791609573658189719 n^{4}+135926787300484940825 n^{3}+2478803467226213273093 n^{2}+20063731171557967684621 n +60812053820266078490706\right) a \! \left(n +11\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{4 \left(17387315312179207913 n^{4}+904216760661564479254 n^{3}+17619141743776933058875 n^{2}+152454422118878431745414 n +494238600179903483492592\right) a \! \left(n +12\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{8 \left(10404967866607148371 n^{4}+569509010828303116405 n^{3}+11678782662230086519943 n^{2}+106348703077903149567611 n +362855080233445811924778\right) a \! \left(n +13\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(31734775106786906287 n^{4}+1749743149195419570118 n^{3}+35934435174524407653821 n^{2}+325504760453319636351014 n +1096116616313714630269296\right) a \! \left(n +14\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{8 \left(838820623429852751 n^{4}+126278325181210458970 n^{3}+4735899652784662549246 n^{2}+68245585565829721330145 n +341812200042095939599620\right) a \! \left(n +15\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{8 \left(15197021134378574734 n^{4}+1137794634904024824994 n^{3}+31603043438226398820683 n^{2}+386588345168371562469713 n +1759454158114665172179726\right) a \! \left(n +16\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(122816098210085221723 n^{4}+9331194013525163750970 n^{3}+264715349328901879711121 n^{2}+3324389778504423981946914 n +15598115908914820999791528\right) a \! \left(n +17\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(327914987019101998473 n^{4}+25900145704973716726322 n^{3}+765103798569356068369227 n^{2}+10019724142748887860867346 n +49087378300489152585413304\right) a \! \left(n +18\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(165752897093945872091 n^{4}+13715700460522495681992 n^{3}+424672505418211911169729 n^{2}+5831693886097097038847976 n +29970107321962481615068668\right) a \! \left(n +19\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(256585241086147242151 n^{4}+22431522365657882376174 n^{3}+733361258964664187594057 n^{2}+10628496827312880516312186 n +57623650222736236150405416\right) a \! \left(n +20\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(138916148957156323235 n^{4}+13138322919933233564466 n^{3}+462738040353604669596817 n^{2}+7199770625025068208817938 n +41784819735652601072732712\right) a \! \left(n +21\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{2 \left(13091135735209972807 n^{4}+1675469681624634346676 n^{3}+73483346020344853504229 n^{2}+1358092131277667440702012 n +9085525953681097899376680\right) a \! \left(n +22\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{3 \left(16127260067929406389 n^{4}+1261675131591453589344 n^{3}+35525072691960326648999 n^{2}+417032649976366853527992 n +1637025861533872830863756\right) a \! \left(n +23\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(38680357766949485170 n^{4}+3547008443995735332491 n^{3}+121571849398611394919945 n^{2}+1845861222526589109073546 n +10476500628629104807907664\right) a \! \left(n +24\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{2 \left(36460078588255178929 n^{4}+3607194871364283273430 n^{3}+133952086787361272789945 n^{2}+2213769220589877048346376 n +13744966601511699330353304\right) a \! \left(n +25\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{3 \left(17662245910564603639 n^{4}+1858094828421181480796 n^{3}+73458289059037160196735 n^{2}+1293861726344971132584642 n +8569566953290322789331784\right) a \! \left(n +26\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(31060890694619400413 n^{4}+3450218189966555609198 n^{3}+144048333851470659083047 n^{2}+2679525340569402401041750 n +18740162722787580820097616\right) a \! \left(n +27\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(13692380689815037689 n^{4}+1598182812854991669353 n^{3}+70106498227344212602356 n^{2}+1369901408174483498288074 n +10061255759176750190720004\right) a \! \left(n +28\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(2657247316796376487 n^{4}+330914954087472934202 n^{3}+15495069199396871421857 n^{2}+323281082182026261553990 n +2535185828406366901612932\right) a \! \left(n +29\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(2567281240368644117 n^{4}+310825403480106935975 n^{3}+14080441633071250524724 n^{2}+282787007422540666378630 n +2123961497418970923501468\right) a \! \left(n +30\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(3632318161790348001 n^{4}+460343215828375705892 n^{3}+21865567223657779338837 n^{2}+461315648868195487192522 n +3647527456403097207271512\right) a \! \left(n +31\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(2635984997078070559 n^{4}+345246991732254809294 n^{3}+16950192676402799382233 n^{2}+369711114451167232379014 n +3022779010891746206734488\right) a \! \left(n +32\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{3 \left(433275303191565294 n^{4}+58530580396848108597 n^{3}+2963902167476333396343 n^{2}+66679977317024963127338 n +562332934699306837281848\right) a \! \left(n +33\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(434659762373805766 n^{4}+60577725387573921843 n^{3}+3164641890208823461667 n^{2}+73447406979567317917044 n +638981542784815495281960\right) a \! \left(n +34\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(81308773524044975 n^{4}+11732948578267950488 n^{3}+634574440509680145043 n^{2}+15246376583311549701364 n +137305893064299350601504\right) a \! \left(n +35\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(2529921999586388 n^{4}+340909442931676659 n^{3}+17058010578625916803 n^{2}+374655303055059877764 n +3036535198418514495120\right) a \! \left(n +36\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(6111970606153223 n^{4}+929020420137259507 n^{3}+52905638661191203756 n^{2}+1337828377311062320628 n +12674711750395260810816\right) a \! \left(n +37\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(758227674787723 n^{4}+122834354133113059 n^{3}+7439383274185675568 n^{2}+199698563399698488644 n +2005249905544050300936\right) a \! \left(n +38\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(151868234327591 n^{4}+29586700413577648 n^{3}+2085449562096138013 n^{2}+63707543987369969600 n +716355695245479576300\right) a \! \left(n +39\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(24071763905554 n^{4}+4770275801595469 n^{3}+343515615807315443 n^{2}+10749248066119170290 n +124021258153391319972\right) a \! \left(n +40\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{2 \left(16911876835548 n^{4}+3003721221379049 n^{3}+199535056042934223 n^{2}+5877315958778532232 n +64782578144726655036\right) a \! \left(n +41\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(9761371151215 n^{4}+1722657594685938 n^{3}+114013613547792473 n^{2}+3353979092367671346 n +37001063980931876832\right) a \! \left(n +42\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(865942916165 n^{4}+154748455929630 n^{3}+10387952163979015 n^{2}+310400141944998282 n +3482976216466678728\right) a \! \left(n +43\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(167951143909 n^{4}+32015543547505 n^{3}+2278292332287764 n^{2}+71775825380327762 n +845084226369332100\right) a \! \left(n +44\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(34827814457 n^{4}+7347923497500 n^{3}+571393492775953 n^{2}+19484569604806230 n +246499427108345940\right) a \! \left(n +45\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(9973721551 n^{4}+1640039780157 n^{3}+99109295893364 n^{2}+2589647814333498 n +24386243883474300\right) a \! \left(n +46\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(1070463895 n^{4}+198852761428 n^{3}+13837735335239 n^{2}+427491617969810 n +4946512402382856\right) a \! \left(n +47\right)}{625 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(178351355 n^{4}+34501608694 n^{3}+2502525293461 n^{2}+80664143069870 n +974899608407520\right) a \! \left(n +48\right)}{625 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{3 \left(212330 n^{4}+51824907 n^{3}+4595697593 n^{2}+177057695390 n +2514697155600\right) a \! \left(n +49\right)}{625 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{3 \left(1475307 n^{4}+296729439 n^{3}+22375510328 n^{2}+749726661246 n +9418065561000\right) a \! \left(n +50\right)}{625 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{512 \left(n +3\right) \left(4911330196364 n^{3}+70341931089269 n^{2}+333734115676060 n +523804197516352\right) a \! \left(n +3\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2048 \left(n +3\right) \left(n +2\right) \left(141907162472 n^{2}+1229372938639 n +2648266363268\right) a \! \left(n +2\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}, \quad n \geq 53\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 110\)
\(\displaystyle a \! \left(6\right) = 544\)
\(\displaystyle a \! \left(7\right) = 2814\)
\(\displaystyle a \! \left(8\right) = 15014\)
\(\displaystyle a \! \left(9\right) = 81982\)
\(\displaystyle a \! \left(10\right) = 455894\)
\(\displaystyle a \! \left(11\right) = 2573022\)
\(\displaystyle a \! \left(12\right) = 14701354\)
\(\displaystyle a \! \left(13\right) = 84871918\)
\(\displaystyle a \! \left(14\right) = 494316702\)
\(\displaystyle a \! \left(15\right) = 2901059978\)
\(\displaystyle a \! \left(16\right) = 17139336394\)
\(\displaystyle a \! \left(17\right) = 101851923374\)
\(\displaystyle a \! \left(18\right) = 608406199902\)
\(\displaystyle a \! \left(19\right) = 3651110194354\)
\(\displaystyle a \! \left(20\right) = 22001788156210\)
\(\displaystyle a \! \left(21\right) = 133081427189222\)
\(\displaystyle a \! \left(22\right) = 807704347570678\)
\(\displaystyle a \! \left(23\right) = 4917360106158202\)
\(\displaystyle a \! \left(24\right) = 30022165036930698\)
\(\displaystyle a \! \left(25\right) = 183773180635729838\)
\(\displaystyle a \! \left(26\right) = 1127622885663073662\)
\(\displaystyle a \! \left(27\right) = 6934397783301056770\)
\(\displaystyle a \! \left(28\right) = 42731366510247561954\)
\(\displaystyle a \! \left(29\right) = 263824647059599069574\)
\(\displaystyle a \! \left(30\right) = 1631767403417108951814\)
\(\displaystyle a \! \left(31\right) = 10109384568340727836234\)
\(\displaystyle a \! \left(32\right) = 62729071246726359088090\)
\(\displaystyle a \! \left(33\right) = 389806433720715603160158\)
\(\displaystyle a \! \left(34\right) = 2425644833897932329065838\)
\(\displaystyle a \! \left(35\right) = 15113631742148392087327378\)
\(\displaystyle a \! \left(36\right) = 94284902341556014933816850\)
\(\displaystyle a \! \left(37\right) = 588868133361715365125620214\)
\(\displaystyle a \! \left(38\right) = 3681880926072536273923978838\)
\(\displaystyle a \! \left(39\right) = 23044777003175577310852255450\)
\(\displaystyle a \! \left(40\right) = 144378836234286920372919379466\)
\(\displaystyle a \! \left(41\right) = 905402482249504431366604747854\)
\(\displaystyle a \! \left(42\right) = 5682864878739801813357413075134\)
\(\displaystyle a \! \left(43\right) = 35699510062198123315151142313378\)
\(\displaystyle a \! \left(44\right) = 224444768335622620603114016499522\)
\(\displaystyle a \! \left(45\right) = 1412190201972625298892619316422118\)
\(\displaystyle a \! \left(46\right) = 8891982356014798595843457306523942\)
\(\displaystyle a \! \left(47\right) = 56028868212814763851004223148289130\)
\(\displaystyle a \! \left(48\right) = 353280777686400836323157874709894202\)
\(\displaystyle a \! \left(49\right) = 2229004312186824174560207630375785534\)
\(\displaystyle a \! \left(50\right) = 14072560730489362668118100304584882446\)
\(\displaystyle a \! \left(51\right) = 88898807148911691278014610490609440690\)
\(\displaystyle a \! \left(52\right) = 561913152555701557453262540645872882290\)
\(\displaystyle a \! \left(n +53\right) = -\frac{684571623424 n \left(n +3\right) \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{671744 \left(31367421 n +123968152\right) \left(n +3\right) \left(n +2\right) \left(n +1\right) a \! \left(n +1\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(142325 n^{3}+21853936 n^{2}+1118300001 n +19070710440\right) a \! \left(n +51\right)}{125 \left(n +54\right) \left(n +53\right) \left(2 n +109\right)}+\frac{\left(1929 n^{2}+201704 n +5269890\right) a \! \left(n +52\right)}{25 \left(2 n +109\right) \left(n +54\right)}-\frac{256 \left(61750953173387 n^{4}+1301222274794262 n^{3}+10203378066585886 n^{2}+35242192236622737 n +45184497177052260\right) a \! \left(n +4\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{128 \left(616254820865543 n^{4}+15407217124459762 n^{3}+143794240050268804 n^{2}+593186574910597037 n +911814806572939032\right) a \! \left(n +5\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{64 \left(5163068112875357 n^{4}+150241148039818720 n^{3}+1634273609803891972 n^{2}+7869990955134689459 n +14146758080334252378\right) a \! \left(n +6\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{32 \left(37365992537243693 n^{4}+1242830512620975881 n^{3}+15460069981490023243 n^{2}+85194986825310888553 n +175395080679324695166\right) a \! \left(n +7\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{32 \left(117402168039830971 n^{4}+4386231178759590298 n^{3}+61300470875537879831 n^{2}+379673530848347682362 n +879020130259163162988\right) a \! \left(n +8\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{16 \left(635004810060654751 n^{4}+26237347800960043901 n^{3}+405644950057376848463 n^{2}+2780569185206532680563 n +7128568563302630455290\right) a \! \left(n +9\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{16 \left(1457584839532261818 n^{4}+65741172673748813335 n^{3}+1109971170358619567397 n^{2}+8313333085661486372954 n +23301336196500640442304\right) a \! \left(n +10\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{16 \left(2791609573658189719 n^{4}+135926787300484940825 n^{3}+2478803467226213273093 n^{2}+20063731171557967684621 n +60812053820266078490706\right) a \! \left(n +11\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{4 \left(17387315312179207913 n^{4}+904216760661564479254 n^{3}+17619141743776933058875 n^{2}+152454422118878431745414 n +494238600179903483492592\right) a \! \left(n +12\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{8 \left(10404967866607148371 n^{4}+569509010828303116405 n^{3}+11678782662230086519943 n^{2}+106348703077903149567611 n +362855080233445811924778\right) a \! \left(n +13\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(31734775106786906287 n^{4}+1749743149195419570118 n^{3}+35934435174524407653821 n^{2}+325504760453319636351014 n +1096116616313714630269296\right) a \! \left(n +14\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{8 \left(838820623429852751 n^{4}+126278325181210458970 n^{3}+4735899652784662549246 n^{2}+68245585565829721330145 n +341812200042095939599620\right) a \! \left(n +15\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{8 \left(15197021134378574734 n^{4}+1137794634904024824994 n^{3}+31603043438226398820683 n^{2}+386588345168371562469713 n +1759454158114665172179726\right) a \! \left(n +16\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(122816098210085221723 n^{4}+9331194013525163750970 n^{3}+264715349328901879711121 n^{2}+3324389778504423981946914 n +15598115908914820999791528\right) a \! \left(n +17\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(327914987019101998473 n^{4}+25900145704973716726322 n^{3}+765103798569356068369227 n^{2}+10019724142748887860867346 n +49087378300489152585413304\right) a \! \left(n +18\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(165752897093945872091 n^{4}+13715700460522495681992 n^{3}+424672505418211911169729 n^{2}+5831693886097097038847976 n +29970107321962481615068668\right) a \! \left(n +19\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(256585241086147242151 n^{4}+22431522365657882376174 n^{3}+733361258964664187594057 n^{2}+10628496827312880516312186 n +57623650222736236150405416\right) a \! \left(n +20\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(138916148957156323235 n^{4}+13138322919933233564466 n^{3}+462738040353604669596817 n^{2}+7199770625025068208817938 n +41784819735652601072732712\right) a \! \left(n +21\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{2 \left(13091135735209972807 n^{4}+1675469681624634346676 n^{3}+73483346020344853504229 n^{2}+1358092131277667440702012 n +9085525953681097899376680\right) a \! \left(n +22\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{3 \left(16127260067929406389 n^{4}+1261675131591453589344 n^{3}+35525072691960326648999 n^{2}+417032649976366853527992 n +1637025861533872830863756\right) a \! \left(n +23\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(38680357766949485170 n^{4}+3547008443995735332491 n^{3}+121571849398611394919945 n^{2}+1845861222526589109073546 n +10476500628629104807907664\right) a \! \left(n +24\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{2 \left(36460078588255178929 n^{4}+3607194871364283273430 n^{3}+133952086787361272789945 n^{2}+2213769220589877048346376 n +13744966601511699330353304\right) a \! \left(n +25\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{3 \left(17662245910564603639 n^{4}+1858094828421181480796 n^{3}+73458289059037160196735 n^{2}+1293861726344971132584642 n +8569566953290322789331784\right) a \! \left(n +26\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(31060890694619400413 n^{4}+3450218189966555609198 n^{3}+144048333851470659083047 n^{2}+2679525340569402401041750 n +18740162722787580820097616\right) a \! \left(n +27\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(13692380689815037689 n^{4}+1598182812854991669353 n^{3}+70106498227344212602356 n^{2}+1369901408174483498288074 n +10061255759176750190720004\right) a \! \left(n +28\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(2657247316796376487 n^{4}+330914954087472934202 n^{3}+15495069199396871421857 n^{2}+323281082182026261553990 n +2535185828406366901612932\right) a \! \left(n +29\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(2567281240368644117 n^{4}+310825403480106935975 n^{3}+14080441633071250524724 n^{2}+282787007422540666378630 n +2123961497418970923501468\right) a \! \left(n +30\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(3632318161790348001 n^{4}+460343215828375705892 n^{3}+21865567223657779338837 n^{2}+461315648868195487192522 n +3647527456403097207271512\right) a \! \left(n +31\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(2635984997078070559 n^{4}+345246991732254809294 n^{3}+16950192676402799382233 n^{2}+369711114451167232379014 n +3022779010891746206734488\right) a \! \left(n +32\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{3 \left(433275303191565294 n^{4}+58530580396848108597 n^{3}+2963902167476333396343 n^{2}+66679977317024963127338 n +562332934699306837281848\right) a \! \left(n +33\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(434659762373805766 n^{4}+60577725387573921843 n^{3}+3164641890208823461667 n^{2}+73447406979567317917044 n +638981542784815495281960\right) a \! \left(n +34\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(81308773524044975 n^{4}+11732948578267950488 n^{3}+634574440509680145043 n^{2}+15246376583311549701364 n +137305893064299350601504\right) a \! \left(n +35\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(2529921999586388 n^{4}+340909442931676659 n^{3}+17058010578625916803 n^{2}+374655303055059877764 n +3036535198418514495120\right) a \! \left(n +36\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(6111970606153223 n^{4}+929020420137259507 n^{3}+52905638661191203756 n^{2}+1337828377311062320628 n +12674711750395260810816\right) a \! \left(n +37\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(758227674787723 n^{4}+122834354133113059 n^{3}+7439383274185675568 n^{2}+199698563399698488644 n +2005249905544050300936\right) a \! \left(n +38\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(151868234327591 n^{4}+29586700413577648 n^{3}+2085449562096138013 n^{2}+63707543987369969600 n +716355695245479576300\right) a \! \left(n +39\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2 \left(24071763905554 n^{4}+4770275801595469 n^{3}+343515615807315443 n^{2}+10749248066119170290 n +124021258153391319972\right) a \! \left(n +40\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{2 \left(16911876835548 n^{4}+3003721221379049 n^{3}+199535056042934223 n^{2}+5877315958778532232 n +64782578144726655036\right) a \! \left(n +41\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(9761371151215 n^{4}+1722657594685938 n^{3}+114013613547792473 n^{2}+3353979092367671346 n +37001063980931876832\right) a \! \left(n +42\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(865942916165 n^{4}+154748455929630 n^{3}+10387952163979015 n^{2}+310400141944998282 n +3482976216466678728\right) a \! \left(n +43\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(167951143909 n^{4}+32015543547505 n^{3}+2278292332287764 n^{2}+71775825380327762 n +845084226369332100\right) a \! \left(n +44\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(34827814457 n^{4}+7347923497500 n^{3}+571393492775953 n^{2}+19484569604806230 n +246499427108345940\right) a \! \left(n +45\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(9973721551 n^{4}+1640039780157 n^{3}+99109295893364 n^{2}+2589647814333498 n +24386243883474300\right) a \! \left(n +46\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{\left(1070463895 n^{4}+198852761428 n^{3}+13837735335239 n^{2}+427491617969810 n +4946512402382856\right) a \! \left(n +47\right)}{625 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{\left(178351355 n^{4}+34501608694 n^{3}+2502525293461 n^{2}+80664143069870 n +974899608407520\right) a \! \left(n +48\right)}{625 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{3 \left(212330 n^{4}+51824907 n^{3}+4595697593 n^{2}+177057695390 n +2514697155600\right) a \! \left(n +49\right)}{625 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{3 \left(1475307 n^{4}+296729439 n^{3}+22375510328 n^{2}+749726661246 n +9418065561000\right) a \! \left(n +50\right)}{625 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}+\frac{512 \left(n +3\right) \left(4911330196364 n^{3}+70341931089269 n^{2}+333734115676060 n +523804197516352\right) a \! \left(n +3\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}-\frac{2048 \left(n +3\right) \left(n +2\right) \left(141907162472 n^{2}+1229372938639 n +2648266363268\right) a \! \left(n +2\right)}{3125 \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(2 n +109\right)}, \quad n \geq 53\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 56 rules.
Found on January 25, 2022.Finding the specification took 1374 seconds.
Copy 56 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_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{23}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{14}\! \left(x \right) &= -F_{50}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{23}\! \left(x \right) &= x\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{23}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= \frac{F_{27}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{2}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{23}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{23}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{39}\! \left(x \right) &= -F_{43}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= \frac{F_{41}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{0}\! \left(x \right) F_{23}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{21}\! \left(x \right) F_{23}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= -F_{50}\! \left(x \right)+F_{8}\! \left(x \right)\\
\end{align*}\)