Av(13452, 13524, 13542, 31452, 31524, 31542, 34152)
Counting Sequence
1, 1, 2, 6, 24, 113, 580, 3127, 17398, 98984, 572823, 3360355, 19935519, 119394778, 720894171, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(4 x^{2}+2 x -1\right) \left(x -1\right)^{2} F \left(x
\right)^{4}-\left(x -1\right) \left(12 x^{3}-17 x^{2}-2 x +2\right) F \left(x
\right)^{3}+\left(8 x^{4}-13 x^{3}-18 x^{2}+32 x -8\right) F \left(x
\right)^{2}-\left(-2+x \right) \left(12 x^{2}-20 x +5\right) F \! \left(x \right)+2 \left(-2+x \right) \left(2 x^{2}-4 x +1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 580\)
\(\displaystyle a(7) = 3127\)
\(\displaystyle a(8) = 17398\)
\(\displaystyle a(9) = 98984\)
\(\displaystyle a(10) = 572823\)
\(\displaystyle a(11) = 3360355\)
\(\displaystyle a(12) = 19935519\)
\(\displaystyle a(13) = 119394778\)
\(\displaystyle a(14) = 720894171\)
\(\displaystyle a(15) = 4383495872\)
\(\displaystyle a(16) = 26819792074\)
\(\displaystyle a(17) = 164992863443\)
\(\displaystyle a(18) = 1019976977915\)
\(\displaystyle a(19) = 6333035600619\)
\(\displaystyle a(20) = 39476901695957\)
\(\displaystyle a(21) = 246958060310719\)
\(\displaystyle a(22) = 1549934430709729\)
\(\displaystyle a(23) = 9756458799133413\)
\(\displaystyle a(24) = 61581969546515962\)
\(\displaystyle a(25) = 389676011595264549\)
\(\displaystyle a(26) = 2471493290055118424\)
\(\displaystyle a(27) = 15708940959882182770\)
\(\displaystyle a(28) = 100046052223484758616\)
\(\displaystyle a(29) = 638350521182417754797\)
\(\displaystyle a(30) = 4080103712648456239394\)
\(\displaystyle a(31) = 26120864887309751706807\)
\(\displaystyle a(32) = 167480559865141965508982\)
\(\displaystyle a(33) = 1075379739329261662952164\)
\(\displaystyle a(34) = 6914224565290645208777329\)
\(\displaystyle a(35) = 44511885202312903000015275\)
\(\displaystyle a(36) = 286898860916948531519820219\)
\(\displaystyle a(37) = 1851287649091624836490888232\)
\(\displaystyle a(38) = 11958737978907289158494454581\)
\(\displaystyle a(39) = 77328443590817506547625458678\)
\(\displaystyle a(40) = 500510912167362420269619763133\)
\(\displaystyle a(41) = 3242557583847504776513701457713\)
\(\displaystyle a{\left(n + 42 \right)} = \frac{2918400000 \left(n + 2\right) \left(2 n + 1\right) \left(2 n + 3\right) a{\left(n \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{9 \left(17 n + 683\right) a{\left(n + 41 \right)}}{2 \left(n + 43\right)} - \frac{\left(10841 n^{2} + 860975 n + 17092860\right) a{\left(n + 40 \right)}}{4 \left(n + 42\right) \left(n + 43\right)} - \frac{29184000 \left(2 n + 3\right) \left(5837 n^{2} + 31948 n + 43121\right) a{\left(n + 1 \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(464899 n^{3} + 54758742 n^{2} + 2149664651 n + 28126135404\right) a{\left(n + 39 \right)}}{8 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(13025483 n^{3} + 1503745248 n^{2} + 57857826679 n + 741917431470\right) a{\left(n + 38 \right)}}{16 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(227253541 n^{3} + 25923685422 n^{2} + 985197121967 n + 12473551625958\right) a{\left(n + 37 \right)}}{32 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(1324254251 n^{3} + 162896417556 n^{2} + 6588268615753 n + 87838497063072\right) a{\left(n + 36 \right)}}{64 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{3200 \left(1467324814 n^{3} + 14579197638 n^{2} + 47302397807 n + 50058546060\right) a{\left(n + 2 \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(58576407631 n^{3} + 5698258815102 n^{2} + 183363010690355 n + 1949055598429920\right) a{\left(n + 35 \right)}}{128 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{480 \left(84615878662 n^{3} + 1090611219197 n^{2} + 4620467773613 n + 6438628721568\right) a{\left(n + 3 \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(1203060539992 n^{3} + 118478598069903 n^{2} + 3881625707288177 n + 42301705991279460\right) a{\left(n + 34 \right)}}{128 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{10 \left(24701049706913 n^{3} + 391076856829671 n^{2} + 2040976302251410 n + 3515305574460696\right) a{\left(n + 4 \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(27148350124093 n^{3} + 2621069885974878 n^{2} + 84249170884165589 n + 901551083521177008\right) a{\left(n + 33 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{3 \left(75257506007428 n^{3} + 7075426434032831 n^{2} + 221518385534534353 n + 2309488870523174768\right) a{\left(n + 32 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{5 \left(223990653261883 n^{3} + 4202897091370239 n^{2} + 26024857172055212 n + 53252972027173188\right) a{\left(n + 5 \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{15 \left(521241435892999 n^{3} + 11298423971028131 n^{2} + 80826214193543352 n + 191101779468922492\right) a{\left(n + 6 \right)}}{2 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(1500025176740003 n^{3} + 136894291355725029 n^{2} + 4160627605271346472 n + 42112999900893644148\right) a{\left(n + 31 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(8251633892449861 n^{3} + 729346064531128179 n^{2} + 21468895850853209006 n + 210457739022792215628\right) a{\left(n + 30 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(12772893909386545 n^{3} + 1091252707329362969 n^{2} + 31046300494637265516 n + 294128269108461180140\right) a{\left(n + 29 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(14210002236495701 n^{3} + 348887728966247147 n^{2} + 2824555177776138942 n + 7551223670005359220\right) a{\left(n + 7 \right)}}{4 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{3 \left(60270764331515475 n^{3} + 1648316318708856137 n^{2} + 14830571345409299158 n + 43960000078852199940\right) a{\left(n + 8 \right)}}{8 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(76416657766936911 n^{3} - 1144813465611684025 n^{2} - 79377689277446652552 n - 648697125524895424436\right) a{\left(n + 11 \right)}}{64 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{3 \left(197448166361766480 n^{3} + 6238341114049340043 n^{2} + 63137480269135675743 n + 203345782167723630320\right) a{\left(n + 10 \right)}}{16 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(303883774231233805 n^{3} + 25044560306065831248 n^{2} + 687228113529633932615 n + 6278525732422025734020\right) a{\left(n + 28 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{9 \left(338248211082511082 n^{3} + 25759502411652131899 n^{2} + 652717452771693973167 n + 5502393811222308097308\right) a{\left(n + 26 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(345257477993130252 n^{3} + 27387999701866046643 n^{2} + 723180683795386490125 n + 6355918225908805491864\right) a{\left(n + 27 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(575006556226792459 n^{3} + 17207554984351263231 n^{2} + 168473146288267795832 n + 540067620536987988600\right) a{\left(n + 9 \right)}}{16 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(2571540551000817993 n^{3} + 187397326650668743526 n^{2} + 4540594284663509350991 n + 36571980266960204034768\right) a{\left(n + 25 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(2624456810635294552 n^{3} + 116480323364816064561 n^{2} + 1744568450893818823784 n + 8742687541572233502222\right) a{\left(n + 12 \right)}}{32 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(8396123624358852397 n^{3} + 582887480247032019642 n^{2} + 13437798610554075641720 n + 102829035658373592854640\right) a{\left(n + 24 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{3 \left(8567530415680937803 n^{3} + 392872759644333982897 n^{2} + 5211134218165823889722 n + 15530797486984765514188\right) a{\left(n + 20 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(12497602668145275908 n^{3} - 398563930563484394319 n^{2} - 30810686215543483635191 n - 356676340287998257857444\right) a{\left(n + 19 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(19053127315495184496 n^{3} + 1529583058608077167399 n^{2} + 37547066456406839593917 n + 291821804118548181303644\right) a{\left(n + 18 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(27155365408029953486 n^{3} + 1249863027889127868081 n^{2} + 19289978429288169533923 n + 99501229130592542661240\right) a{\left(n + 13 \right)}}{128 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(31101983011309607099 n^{3} + 2043110475127566575067 n^{2} + 44460975099707972891956 n + 320184712381498817878272\right) a{\left(n + 23 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(48064138372727441954 n^{3} + 2951854940103539705685 n^{2} + 59723481461861450047027 n + 396909313377476966311572\right) a{\left(n + 22 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(58719030854975530289 n^{3} + 2880426183096232477239 n^{2} + 47189292290406384867786 n + 257758075291581625555760\right) a{\left(n + 14 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(59303085233011707575 n^{3} + 3312700744632902758881 n^{2} + 59931156205922133595276 n + 346618336612633164172656\right) a{\left(n + 21 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(99773734421571385094 n^{3} + 5714002645036334232660 n^{2} + 108260591852219193162007 n + 679041845732710309833486\right) a{\left(n + 16 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(138500659097221159456 n^{3} + 8849076246166889137917 n^{2} + 184525598462021571911147 n + 1261740151757873814961188\right) a{\left(n + 17 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(213467022030358701911 n^{3} + 11255769644823473950551 n^{2} + 197462429109300780615196 n + 1151772880342606799924460\right) a{\left(n + 15 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)}, \quad n \geq 42\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 580\)
\(\displaystyle a(7) = 3127\)
\(\displaystyle a(8) = 17398\)
\(\displaystyle a(9) = 98984\)
\(\displaystyle a(10) = 572823\)
\(\displaystyle a(11) = 3360355\)
\(\displaystyle a(12) = 19935519\)
\(\displaystyle a(13) = 119394778\)
\(\displaystyle a(14) = 720894171\)
\(\displaystyle a(15) = 4383495872\)
\(\displaystyle a(16) = 26819792074\)
\(\displaystyle a(17) = 164992863443\)
\(\displaystyle a(18) = 1019976977915\)
\(\displaystyle a(19) = 6333035600619\)
\(\displaystyle a(20) = 39476901695957\)
\(\displaystyle a(21) = 246958060310719\)
\(\displaystyle a(22) = 1549934430709729\)
\(\displaystyle a(23) = 9756458799133413\)
\(\displaystyle a(24) = 61581969546515962\)
\(\displaystyle a(25) = 389676011595264549\)
\(\displaystyle a(26) = 2471493290055118424\)
\(\displaystyle a(27) = 15708940959882182770\)
\(\displaystyle a(28) = 100046052223484758616\)
\(\displaystyle a(29) = 638350521182417754797\)
\(\displaystyle a(30) = 4080103712648456239394\)
\(\displaystyle a(31) = 26120864887309751706807\)
\(\displaystyle a(32) = 167480559865141965508982\)
\(\displaystyle a(33) = 1075379739329261662952164\)
\(\displaystyle a(34) = 6914224565290645208777329\)
\(\displaystyle a(35) = 44511885202312903000015275\)
\(\displaystyle a(36) = 286898860916948531519820219\)
\(\displaystyle a(37) = 1851287649091624836490888232\)
\(\displaystyle a(38) = 11958737978907289158494454581\)
\(\displaystyle a(39) = 77328443590817506547625458678\)
\(\displaystyle a(40) = 500510912167362420269619763133\)
\(\displaystyle a(41) = 3242557583847504776513701457713\)
\(\displaystyle a{\left(n + 42 \right)} = \frac{2918400000 \left(n + 2\right) \left(2 n + 1\right) \left(2 n + 3\right) a{\left(n \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{9 \left(17 n + 683\right) a{\left(n + 41 \right)}}{2 \left(n + 43\right)} - \frac{\left(10841 n^{2} + 860975 n + 17092860\right) a{\left(n + 40 \right)}}{4 \left(n + 42\right) \left(n + 43\right)} - \frac{29184000 \left(2 n + 3\right) \left(5837 n^{2} + 31948 n + 43121\right) a{\left(n + 1 \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(464899 n^{3} + 54758742 n^{2} + 2149664651 n + 28126135404\right) a{\left(n + 39 \right)}}{8 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(13025483 n^{3} + 1503745248 n^{2} + 57857826679 n + 741917431470\right) a{\left(n + 38 \right)}}{16 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(227253541 n^{3} + 25923685422 n^{2} + 985197121967 n + 12473551625958\right) a{\left(n + 37 \right)}}{32 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(1324254251 n^{3} + 162896417556 n^{2} + 6588268615753 n + 87838497063072\right) a{\left(n + 36 \right)}}{64 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{3200 \left(1467324814 n^{3} + 14579197638 n^{2} + 47302397807 n + 50058546060\right) a{\left(n + 2 \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(58576407631 n^{3} + 5698258815102 n^{2} + 183363010690355 n + 1949055598429920\right) a{\left(n + 35 \right)}}{128 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{480 \left(84615878662 n^{3} + 1090611219197 n^{2} + 4620467773613 n + 6438628721568\right) a{\left(n + 3 \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(1203060539992 n^{3} + 118478598069903 n^{2} + 3881625707288177 n + 42301705991279460\right) a{\left(n + 34 \right)}}{128 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{10 \left(24701049706913 n^{3} + 391076856829671 n^{2} + 2040976302251410 n + 3515305574460696\right) a{\left(n + 4 \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(27148350124093 n^{3} + 2621069885974878 n^{2} + 84249170884165589 n + 901551083521177008\right) a{\left(n + 33 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{3 \left(75257506007428 n^{3} + 7075426434032831 n^{2} + 221518385534534353 n + 2309488870523174768\right) a{\left(n + 32 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{5 \left(223990653261883 n^{3} + 4202897091370239 n^{2} + 26024857172055212 n + 53252972027173188\right) a{\left(n + 5 \right)}}{\left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{15 \left(521241435892999 n^{3} + 11298423971028131 n^{2} + 80826214193543352 n + 191101779468922492\right) a{\left(n + 6 \right)}}{2 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(1500025176740003 n^{3} + 136894291355725029 n^{2} + 4160627605271346472 n + 42112999900893644148\right) a{\left(n + 31 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(8251633892449861 n^{3} + 729346064531128179 n^{2} + 21468895850853209006 n + 210457739022792215628\right) a{\left(n + 30 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(12772893909386545 n^{3} + 1091252707329362969 n^{2} + 31046300494637265516 n + 294128269108461180140\right) a{\left(n + 29 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(14210002236495701 n^{3} + 348887728966247147 n^{2} + 2824555177776138942 n + 7551223670005359220\right) a{\left(n + 7 \right)}}{4 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{3 \left(60270764331515475 n^{3} + 1648316318708856137 n^{2} + 14830571345409299158 n + 43960000078852199940\right) a{\left(n + 8 \right)}}{8 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(76416657766936911 n^{3} - 1144813465611684025 n^{2} - 79377689277446652552 n - 648697125524895424436\right) a{\left(n + 11 \right)}}{64 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{3 \left(197448166361766480 n^{3} + 6238341114049340043 n^{2} + 63137480269135675743 n + 203345782167723630320\right) a{\left(n + 10 \right)}}{16 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(303883774231233805 n^{3} + 25044560306065831248 n^{2} + 687228113529633932615 n + 6278525732422025734020\right) a{\left(n + 28 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{9 \left(338248211082511082 n^{3} + 25759502411652131899 n^{2} + 652717452771693973167 n + 5502393811222308097308\right) a{\left(n + 26 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(345257477993130252 n^{3} + 27387999701866046643 n^{2} + 723180683795386490125 n + 6355918225908805491864\right) a{\left(n + 27 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(575006556226792459 n^{3} + 17207554984351263231 n^{2} + 168473146288267795832 n + 540067620536987988600\right) a{\left(n + 9 \right)}}{16 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(2571540551000817993 n^{3} + 187397326650668743526 n^{2} + 4540594284663509350991 n + 36571980266960204034768\right) a{\left(n + 25 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(2624456810635294552 n^{3} + 116480323364816064561 n^{2} + 1744568450893818823784 n + 8742687541572233502222\right) a{\left(n + 12 \right)}}{32 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(8396123624358852397 n^{3} + 582887480247032019642 n^{2} + 13437798610554075641720 n + 102829035658373592854640\right) a{\left(n + 24 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{3 \left(8567530415680937803 n^{3} + 392872759644333982897 n^{2} + 5211134218165823889722 n + 15530797486984765514188\right) a{\left(n + 20 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(12497602668145275908 n^{3} - 398563930563484394319 n^{2} - 30810686215543483635191 n - 356676340287998257857444\right) a{\left(n + 19 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(19053127315495184496 n^{3} + 1529583058608077167399 n^{2} + 37547066456406839593917 n + 291821804118548181303644\right) a{\left(n + 18 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(27155365408029953486 n^{3} + 1249863027889127868081 n^{2} + 19289978429288169533923 n + 99501229130592542661240\right) a{\left(n + 13 \right)}}{128 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(31101983011309607099 n^{3} + 2043110475127566575067 n^{2} + 44460975099707972891956 n + 320184712381498817878272\right) a{\left(n + 23 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(48064138372727441954 n^{3} + 2951854940103539705685 n^{2} + 59723481461861450047027 n + 396909313377476966311572\right) a{\left(n + 22 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{3 \left(58719030854975530289 n^{3} + 2880426183096232477239 n^{2} + 47189292290406384867786 n + 257758075291581625555760\right) a{\left(n + 14 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(59303085233011707575 n^{3} + 3312700744632902758881 n^{2} + 59931156205922133595276 n + 346618336612633164172656\right) a{\left(n + 21 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} - \frac{\left(99773734421571385094 n^{3} + 5714002645036334232660 n^{2} + 108260591852219193162007 n + 679041845732710309833486\right) a{\left(n + 16 \right)}}{256 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(138500659097221159456 n^{3} + 8849076246166889137917 n^{2} + 184525598462021571911147 n + 1261740151757873814961188\right) a{\left(n + 17 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)} + \frac{\left(213467022030358701911 n^{3} + 11255769644823473950551 n^{2} + 197462429109300780615196 n + 1151772880342606799924460\right) a{\left(n + 15 \right)}}{512 \left(n + 41\right) \left(n + 42\right) \left(n + 43\right)}, \quad n \geq 42\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 128 rules.
Finding the specification took 7629 seconds.
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\(\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_{20}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \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_{21}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= \frac{F_{19}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{20}\! \left(x \right) &= x\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= \frac{F_{28}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{20}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{0}\! \left(x \right) F_{20}\! \left(x \right) F_{36}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{36}\! \left(x \right) &= -F_{65}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= \frac{F_{38}\! \left(x \right)}{F_{20}\! \left(x \right) F_{41}\! \left(x \right) F_{58}\! \left(x \right)}\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{20}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{48}\! \left(x \right) &= 0\\
F_{49}\! \left(x \right) &= F_{20}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{20}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{20}\! \left(x \right) F_{41}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{2}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{20}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{20}\! \left(x \right) F_{58}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{0}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{20}\! \left(x \right) F_{37}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{68}\! \left(x \right) &= \frac{F_{69}\! \left(x \right)}{F_{20}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= -F_{4}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= -F_{124}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{75}\! \left(x \right) &= x^{2} F_{75} \left(x \right)^{2}-2 x F_{75} \left(x \right)^{2}+F_{75}\! \left(x \right) x +2 F_{75}\! \left(x \right)-1\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{20}\! \left(x \right) F_{75}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{20}\! \left(x \right) F_{75}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{20}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= \frac{F_{86}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= -F_{92}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= \frac{F_{89}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= -F_{91}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{91}\! \left(x \right) &= x^{2} F_{91} \left(x \right)^{2}+2 x^{2} F_{91}\! \left(x \right)-2 x F_{91} \left(x \right)^{2}+x^{2}-3 x F_{91}\! \left(x \right)-x +2 F_{91}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{20}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{20}\! \left(x \right) F_{68}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= \frac{F_{99}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{99}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{102}\! \left(x \right) &= \frac{F_{103}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{105}\! \left(x \right) &= \frac{F_{106}\! \left(x \right)}{F_{20}\! \left(x \right) F_{58}\! \left(x \right)}\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\
F_{107}\! \left(x \right) &= -F_{115}\! \left(x \right)+F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= \frac{F_{109}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= -F_{114}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{20}\! \left(x \right) F_{5}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{2}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{116}\! \left(x \right) &= \frac{F_{117}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{118}\! \left(x \right) &= -F_{110}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= -F_{120}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{0}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{121}\! \left(x \right) &= -F_{83}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= \frac{F_{123}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{123}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{127}\! \left(x \right) &= -F_{111}\! \left(x \right)+F_{121}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 129 rules.
Finding the specification took 3194 seconds.
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\(\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_{20}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \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_{21}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= \frac{F_{19}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{20}\! \left(x \right) &= x\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= x^{2} F_{27} \left(x \right)^{2}+2 x^{2} F_{27}\! \left(x \right)-2 x F_{27} \left(x \right)^{2}+x^{2}-3 x F_{27}\! \left(x \right)-x +2 F_{27}\! \left(x \right)\\
F_{28}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= -F_{32}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= \frac{F_{31}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{32}\! \left(x \right) &= x^{2} F_{32} \left(x \right)^{2}-2 x F_{32} \left(x \right)^{2}+F_{32}\! \left(x \right) x +2 F_{32}\! \left(x \right)-1\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{20}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{0}\! \left(x \right) F_{20}\! \left(x \right) F_{40}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{40}\! \left(x \right) &= -F_{69}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{20}\! \left(x \right) F_{45}\! \left(x \right) F_{62}\! \left(x \right)}\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{2}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{20}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{52}\! \left(x \right) &= 0\\
F_{53}\! \left(x \right) &= F_{20}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{20}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{20}\! \left(x \right) F_{45}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{2}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{20}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{20}\! \left(x \right) F_{62}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{0}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{20}\! \left(x \right) F_{41}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{72}\! \left(x \right) &= \frac{F_{73}\! \left(x \right)}{F_{20}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= -F_{4}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= -F_{125}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{20}\! \left(x \right) F_{32}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{20}\! \left(x \right) F_{32}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{20}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= \frac{F_{89}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= -F_{93}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= \frac{F_{92}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{92}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{20}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{20}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= \frac{F_{100}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{20}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{102}\! \left(x \right) &= \frac{F_{103}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{103}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{106}\! \left(x \right) &= \frac{F_{107}\! \left(x \right)}{F_{20}\! \left(x \right) F_{62}\! \left(x \right)}\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= -F_{116}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{109}\! \left(x \right) &= \frac{F_{110}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= -F_{115}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{20}\! \left(x \right) F_{5}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{2}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{117}\! \left(x \right) &= \frac{F_{118}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= -F_{111}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= -F_{121}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{0}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{122}\! \left(x \right) &= -F_{86}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= \frac{F_{124}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{124}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{128}\! \left(x \right) &= -F_{112}\! \left(x \right)+F_{122}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 128 rules.
Finding the specification took 7629 seconds.
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Copy 128 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_{20}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \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_{21}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= \frac{F_{19}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{20}\! \left(x \right) &= x\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= \frac{F_{28}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{20}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{0}\! \left(x \right) F_{20}\! \left(x \right) F_{36}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{36}\! \left(x \right) &= -F_{65}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= \frac{F_{38}\! \left(x \right)}{F_{20}\! \left(x \right) F_{41}\! \left(x \right) F_{58}\! \left(x \right)}\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{20}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{48}\! \left(x \right) &= 0\\
F_{49}\! \left(x \right) &= F_{20}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{20}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{20}\! \left(x \right) F_{41}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{2}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{20}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{20}\! \left(x \right) F_{58}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{0}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{20}\! \left(x \right) F_{37}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{68}\! \left(x \right) &= \frac{F_{69}\! \left(x \right)}{F_{20}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= -F_{4}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= -F_{124}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{75}\! \left(x \right) &= x^{2} F_{75} \left(x \right)^{2}-2 x F_{75} \left(x \right)^{2}+F_{75}\! \left(x \right) x +2 F_{75}\! \left(x \right)-1\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{20}\! \left(x \right) F_{75}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{20}\! \left(x \right) F_{75}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{20}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= \frac{F_{86}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= -F_{92}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= \frac{F_{89}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= -F_{91}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{91}\! \left(x \right) &= x^{2} F_{91} \left(x \right)^{2}+2 x^{2} F_{91}\! \left(x \right)-2 x F_{91} \left(x \right)^{2}+x^{2}-3 x F_{91}\! \left(x \right)-x +2 F_{91}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{20}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{20}\! \left(x \right) F_{68}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= \frac{F_{99}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{99}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{102}\! \left(x \right) &= \frac{F_{103}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{105}\! \left(x \right) &= \frac{F_{106}\! \left(x \right)}{F_{20}\! \left(x \right) F_{58}\! \left(x \right)}\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\
F_{107}\! \left(x \right) &= -F_{115}\! \left(x \right)+F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= \frac{F_{109}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= -F_{114}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{20}\! \left(x \right) F_{5}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{2}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{116}\! \left(x \right) &= \frac{F_{117}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{118}\! \left(x \right) &= -F_{110}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= -F_{120}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{0}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{121}\! \left(x \right) &= -F_{83}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= \frac{F_{123}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{123}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{127}\! \left(x \right) &= -F_{111}\! \left(x \right)+F_{121}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 47 rules.
Finding the specification took 36 seconds.
Copy 47 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_{9}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y_{0}\right) &= F_{6}\! \left(x , y_{0}\right) F_{9}\! \left(x \right)\\
F_{6}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{0}\right)+F_{32}\! \left(x , y_{0}\right)+F_{7}\! \left(x , y_{0}\right)\\
F_{7}\! \left(x , y_{0}\right) &= F_{8}\! \left(x , y_{0}\right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x , y_{0}\right) &= -\frac{-F_{6}\! \left(x , y_{0}\right) y_{0}+F_{6}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , y_{0}\right) F_{9}\! \left(x \right)\\
F_{11}\! \left(x , y_{0}\right) &= -\frac{-F_{12}\! \left(x , y_{0}\right) y_{0}+F_{12}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{12}\! \left(x , y_{0}\right) &= F_{13}\! \left(x , 1, y_{0}\right)\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y_{0}, y_{1}\right)+F_{34}\! \left(x , y_{0}, y_{1}\right)+F_{44}\! \left(x , y_{0}, y_{1}\right)\\
F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0}, y_{1}\right) F_{9}\! \left(x \right)\\
F_{16}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{1}\right)+F_{17}\! \left(x , y_{0}, y_{1}\right)+F_{31}\! \left(x , y_{0}, y_{1}\right)+F_{32}\! \left(x , y_{1}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0}, y_{1}\right) F_{9}\! \left(x \right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{20}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{23}\! \left(x , y_{0}, y_{2}\right)+F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{29}\! \left(x , y_{0}, y_{2}\right)\\
F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{9}\! \left(x \right)\\
F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{20}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{24}\! \left(x , y_{0}, y_{1}\right) F_{26}\! \left(x , y_{0}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{25}\! \left(x , y_{0}, 1\right) y_{0}-F_{25}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{26}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{28}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{0}, y_{2}\right) F_{26}\! \left(x , y_{1}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{26}\! \left(x , y_{1}\right) F_{30}\! \left(x , y_{0}, y_{1}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{14}\! \left(x , 1, y_{1}\right) y_{1}-F_{14}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{26}\! \left(x , y_{1}\right)\\
F_{32}\! \left(x , y_{0}\right) &= F_{26}\! \left(x , y_{0}\right) F_{33}\! \left(x , y_{0}\right)\\
F_{33}\! \left(x , y_{0}\right) &= F_{30}\! \left(x , 1, y_{0}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{26}\! \left(x , y_{1}\right) F_{35}\! \left(x , y_{0}, y_{1}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{36}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{36}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{37}\! \left(x , y_{0}, y_{1}, y_{0} y_{2}\right)\\
F_{37}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{38}\! \left(x , y_{0} y_{1}, y_{2}, y_{1}\right)\\
F_{38}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{29}\! \left(x , y_{0}, y_{2}\right)+F_{39}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{41}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{39}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{9}\! \left(x \right)\\
F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{F_{16}\! \left(x , y_{1}, y_{2}\right) y_{1} y_{2}-F_{16}\! \left(x , \frac{y_{0}}{y_{2}}, y_{2}\right) y_{0}}{-y_{1} y_{2}+y_{0}}\\
F_{41}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{26}\! \left(x , y_{0}\right) F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{-y_{1} y_{2} F_{43}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}, y_{2}\right)+y_{0} F_{43}\! \left(x , y_{0}, \frac{1}{y_{2}}, y_{2}\right)}{-y_{1} y_{2}+y_{0}}\\
F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{38}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\
F_{44}\! \left(x , y_{0}, y_{1}\right) &= F_{29}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\
\end{align*}\)