Av(13452, 13524, 13542, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 31452, 31524, 31542, 34152, 41352, 41523, 41532, 43152)
View Raw Data
Counting Sequence
1, 1, 2, 6, 24, 100, 426, 1855, 8278, 37683, 174230, 815754, 3859996, 18430786, 88693660, ...
Implicit Equation for the Generating Function
\(\displaystyle -x^{4} F \left(x \right)^{5}+x^{3} \left(2 x +3\right) \left(x -1\right) F \left(x \right)^{4}-x^{2} \left(4 x^{3}-2 x +1\right) F \left(x \right)^{3}+x \left(2 x^{4}+2 x^{2}+x -1\right) F \left(x \right)^{2}-\left(x -1\right) \left(x +1\right) F \! \left(x \right)+\left(x -1\right) \left(x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 100\)
\(\displaystyle a \! \left(6\right) = 426\)
\(\displaystyle a \! \left(7\right) = 1855\)
\(\displaystyle a \! \left(8\right) = 8278\)
\(\displaystyle a \! \left(9\right) = 37683\)
\(\displaystyle a \! \left(10\right) = 174230\)
\(\displaystyle a \! \left(11\right) = 815754\)
\(\displaystyle a \! \left(12\right) = 3859996\)
\(\displaystyle a \! \left(13\right) = 18430786\)
\(\displaystyle a \! \left(14\right) = 88693660\)
\(\displaystyle a \! \left(15\right) = 429727342\)
\(\displaystyle a \! \left(16\right) = 2094521248\)
\(\displaystyle a \! \left(17\right) = 10262902034\)
\(\displaystyle a \! \left(18\right) = 50524277614\)
\(\displaystyle a \! \left(19\right) = 249783552965\)
\(\displaystyle a \! \left(20\right) = 1239599877272\)
\(\displaystyle a \! \left(21\right) = 6173031600701\)
\(\displaystyle a \! \left(22\right) = 30837615980210\)
\(\displaystyle a \! \left(23\right) = 154493997874241\)
\(\displaystyle a \! \left(24\right) = 776047860373198\)
\(\displaystyle a \! \left(25\right) = 3907699296142445\)
\(\displaystyle a \! \left(26\right) = 19721022592789142\)
\(\displaystyle a \! \left(27\right) = 99733735318819228\)
\(\displaystyle a \! \left(28\right) = 505353563670636388\)
\(\displaystyle a \! \left(29\right) = 2565263120392117602\)
\(\displaystyle a \! \left(30\right) = 13043683064883500332\)
\(\displaystyle a \! \left(31\right) = 66428374481356825266\)
\(\displaystyle a \! \left(32\right) = 338804942034642975854\)
\(\displaystyle a \! \left(33\right) = 1730414059029770321374\)
\(\displaystyle a \! \left(34\right) = 8849512503161774509414\)
\(\displaystyle a \! \left(35\right) = 45313278342315435920943\)
\(\displaystyle a \! \left(36\right) = 232294491206511113716988\)
\(\displaystyle a \! \left(37\right) = 1192154288197596856783099\)
\(\displaystyle a \! \left(38\right) = 6124647649319499420311524\)
\(\displaystyle a \! \left(39\right) = 31496457572585025658857937\)
\(\displaystyle a \! \left(40\right) = 162126072245995862083544466\)
\(\displaystyle a \! \left(41\right) = 835285036944978253546820487\)
\(\displaystyle a \! \left(42\right) = 4307137939051789710124921760\)
\(\displaystyle a \! \left(43\right) = 22227873853468007314227119275\)
\(\displaystyle a \! \left(44\right) = 114801117917435654094905423660\)
\(\displaystyle a \! \left(45\right) = 593360193898963212582843027411\)
\(\displaystyle a \! \left(46\right) = 3069027709417580430777678914266\)
\(\displaystyle a \! \left(47\right) = 15884744074775149389695826388082\)
\(\displaystyle a \! \left(48\right) = 82270541983826178924443072889100\)
\(\displaystyle a \! \left(49\right) = 426365129025915098109484225378828\)
\(\displaystyle a \! \left(50\right) = 2210962253865199200753212909139962\)
\(\displaystyle a \! \left(51\right) = 11471840246421666713492465730684361\)
\(\displaystyle a \! \left(52\right) = 59556248268648171753247413463653772\)
\(\displaystyle a \! \left(53\right) = 309353447578164125501243343282351049\)
\(\displaystyle a \! \left(54\right) = 1607708793021274855132139413894607094\)
\(\displaystyle a \! \left(55\right) = 8359426580083148997870054781020929138\)
\(\displaystyle a \! \left(56\right) = 43486511623807443459189620631893810836\)
\(\displaystyle a \! \left(57\right) = 226325947865768038149499262639471856530\)
\(\displaystyle a \! \left(58\right) = 1178443876220195195766883927183370982848\)
\(\displaystyle a \! \left(59\right) = 6138633066238254567520947952983617849496\)
\(\displaystyle a \! \left(60\right) = 31990160640927007232459929981614952102576\)
\(\displaystyle a \! \left(61\right) = 166777404254995146624908835396857126991284\)
\(\displaystyle a \! \left(62\right) = 869818007019889229131622371150360701994696\)
\(\displaystyle a \! \left(63\right) = 4538210045476605918566315389282968450917670\)
\(\displaystyle a \! \left(64\right) = 23686488209572326457689640849344603691219456\)
\(\displaystyle a \! \left(65\right) = 123672104476209982059876847692212888048240088\)
\(\displaystyle a \! \left(66\right) = 645941435075040959006320082746952411132323152\)
\(\displaystyle a \! \left(67\right) = 3374895973277901976205198674006419050944873114\)
\(\displaystyle a \! \left(68\right) = 17638807590615415175342888441519349372274186376\)
\(\displaystyle a \! \left(69\right) = 92217969656557308072416339372490902980617868402\)
\(\displaystyle a \! \left(70\right) = 482275812354251301439259905884501765240772688192\)
\(\displaystyle a \! \left(71\right) = 2522930299900289996126424549972978629427250657221\)
\(\displaystyle a \! \left(72\right) = 13202047174432847247513987727908065848729737469466\)
\(\displaystyle a \! \left(73\right) = 69103512216463290588585131347241173259404850308023\)
\(\displaystyle a \! \left(74\right) = 361808254608468329682394044744660786611615799079592\)
\(\displaystyle a \! \left(75\right) = 1894842675154373291918602563047916970447072328359183\)
\(\displaystyle a \! \left(76\right) = 9926156812541193967666467248967139670973375382260316\)
\(\displaystyle a \! \left(n +77\right) = -\frac{2 \left(81812042509 n^{3}+30234831035982 n^{2}+3114995260078187 n +98094179276165058\right) a \! \left(n +74\right)}{61966802493 \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{4 \left(85954063729 n^{2}+12697472411753 n +468571594388070\right) a \! \left(n +75\right)}{20655600831 \left(n +77\right) \left(n +78\right)}+\frac{2 \left(41430097253 n +3253883311109\right) a \! \left(n +76\right)}{20655600831 \left(n +78\right)}+\frac{314703872 n \left(n -1\right) \left(2 n +3\right) \left(n +1\right) a \! \left(n \right)}{61966802493 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{44957696 n \left(2 n +5\right) \left(34 n +67\right) \left(n +1\right) a \! \left(n +1\right)}{20655600831 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{4 \left(514271374692573 n^{4}+142945163107667623 n^{3}+14894302275034051158 n^{2}+689489874885005050598 n +11964749416012322684550\right) a \! \left(n +71\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{2 \left(295468472933543 n^{4}+84489092746998793 n^{3}+9058183379142283213 n^{2}+431536002258819039383 n +7707977456403645781740\right) a \! \left(n +72\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{\left(38045891398849 n^{4}+10961538251787258 n^{3}+1184037813296964347 n^{2}+56829521406629609682 n +1022605884863704359936\right) a \! \left(n +73\right)}{61966802493 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{2 \left(1219515105093148207 n^{4}+305231820169241143013 n^{3}+28632296836087896161555 n^{2}+1193010992510896223322151 n +18629441026706749937211522\right) a \! \left(n +64\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{\left(1145101702507443207 n^{4}+298881439791690073726 n^{3}+29251783334391203303037 n^{2}+1272296943553406072687150 n +20750047108724952796159032\right) a \! \left(n +65\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{2 \left(360167060520562445 n^{4}+93393334878013618753 n^{3}+9078131318779321332967 n^{2}+392038174722363990844259 n +6346315634442915368403480\right) a \! \left(n +66\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{8 \left(4472964666583965 n^{4}+1211994027535604148 n^{3}+123105180518199497964 n^{2}+5555318933310275916962 n +93974904822369066551511\right) a \! \left(n +67\right)}{103278004155 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{2 \left(66559121689193065 n^{4}+17825262028620960869 n^{3}+1789647196447081870049 n^{2}+79833643273609847959999 n +1335066415765180618533954\right) a \! \left(n +68\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{\left(2971052374381043 n^{4}+734922383602734782 n^{3}+67374644660693067709 n^{2}+2705099418261486883210 n +39964603881234150806160\right) a \! \left(n +69\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{4 \left(3468497582028819 n^{4}+958411436606281727 n^{3}+99285480120904562706 n^{2}+4570111921807609328803 n +78865482968404436276385\right) a \! \left(n +70\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{2 \left(715225883333298525 n^{4}-106181660053294345357 n^{3}-31724495610560087551389 n^{2}-2050007681048303223265499 n -40992026643318681037587192\right) a \! \left(n +56\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{\left(161022553070794882753 n^{4}+37897704054722774186586 n^{3}+3342039449075567725215851 n^{2}+130884419637984120241242162 n +1920761186403802194296309424\right) a \! \left(n +57\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{2 \left(384429558612326715 n^{4}-46817399361124002109 n^{3}-15928615571218142764497 n^{2}-1074691625392481396776115 n -22232981715387937476048282\right) a \! \left(n +58\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{4 \left(17950998679903348717 n^{4}+4317583171439080134900 n^{3}+389346299554167245355887 n^{2}+15601338474548450813955408 n +234386655084746329136627124\right) a \! \left(n +59\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{2 \left(3110486653007616871 n^{4}+687644629219028028591 n^{3}+56592246729036317087201 n^{2}+2052109484737024743977433 n +27613866203503081108835232\right) a \! \left(n +60\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{\left(25127486771201565087 n^{4}+6194802854251612383806 n^{3}+572712287927207713105617 n^{2}+23532088701710554621682002 n +362586270707358868054807944\right) a \! \left(n +61\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{4 \left(261681223171785229 n^{4}+62686979662746836411 n^{3}+5624160516928577346341 n^{2}+223957532062041612089521 n +3339467948537309679051354\right) a \! \left(n +62\right)}{61966802493 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{4 \left(1625975063370301764 n^{4}+411904922857419878101 n^{3}+39130382433159761915763 n^{2}+1652150590191482203687874 n +26158662068226834410446608\right) a \! \left(n +63\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{2 \left(1021713660009913698647 n^{4}+205494501034623125185003 n^{3}+15478690286934017041758889 n^{2}+517534554651683592596097737 n +6481131294046303388363474652\right) a \! \left(n +50\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{8 \left(37588914375287235290 n^{4}+9457987922236883017276 n^{3}+860934833275825557742213 n^{2}+33958916892767251967214938 n +492928122985947578489347137\right) a \! \left(n +51\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{2 \left(317735939379832978635 n^{4}+65875003650754020351383 n^{3}+5113018772791400008526559 n^{2}+176091261680078960995782445 n +2270545983089689673890121154\right) a \! \left(n +52\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{\left(439816217799896306575 n^{4}+100767739560834802377094 n^{3}+8610056424661696919279777 n^{2}+325409381657036934866228690 n +4592688142723513672088648928\right) a \! \left(n +53\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{4 \left(27265254123260289893 n^{4}+5691603799010206550965 n^{3}+443763753158598788689948 n^{2}+15311950425180558867031757 n +197217564733144731117074469\right) a \! \left(n +54\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{4 \left(14912816354896871941 n^{4}+3446585551958036509707 n^{3}+298048094582002196787098 n^{2}+11431999237453968027920418 n +164128612053957034937388594\right) a \! \left(n +55\right)}{61966802493 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{4 \left(1105080659794235264069 n^{4}+204579726517574185851359 n^{3}+14156544542092737497939359 n^{2}+434124026184161306864254717 n +4979371542448234130118543552\right) a \! \left(n +44\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{2 \left(5194936908241317007933 n^{4}+922850436594214994853098 n^{3}+61427880798043470608278823 n^{2}+1815764790894423171711486826 n +20110251697566168577971077280\right) a \! \left(n +45\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{4 \left(1476717012099747593051 n^{4}+277848077899851170577993 n^{3}+19576881635958719332664533 n^{2}+612254966207762333275685223 n +7171752627560848898247467352\right) a \! \left(n +46\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{4 \left(1127121577091286043662 n^{4}+204746461266392535000713 n^{3}+13919207148902137228878849 n^{2}+419641665696345030692517346 n +4733038589776669334002705284\right) a \! \left(n +47\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{2 \left(715193086418052684335 n^{4}+139017678334806146122541 n^{3}+10121019253201022774669223 n^{2}+327113634820386710256589207 n +3960339910823643821687112438\right) a \! \left(n +48\right)}{103278004155 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{\left(959015690620757479875 n^{4}+165225948247492304613958 n^{3}+10455584708109045758497425 n^{2}+285847066721375786935892966 n +2812795695945947361710800872\right) a \! \left(n +49\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{32 \left(308934299790025976353 n^{4}+44971517589717830300728 n^{3}+2454130236121875641913173 n^{2}+59501510856249266564619032 n +540802410413975484158394834\right) a \! \left(n +36\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{16 \left(340800452582408119429 n^{4}+53162436883628402863794 n^{3}+3101357653519817501188907 n^{2}+80211350060599625797646526 n +776183843089972063935179028\right) a \! \left(n +37\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{16 \left(709299393549404900037 n^{4}+107470458334876738171345 n^{3}+6104098405823919214458231 n^{2}+154030866294256837884643403 n +1456990209351114885362978400\right) a \! \left(n +38\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{8 \left(1614329459402861889521 n^{4}+257349939909188354709982 n^{3}+15373529461999249148831191 n^{2}+407887088123024864676895298 n +4055547540513899741492895024\right) a \! \left(n +39\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{16 \left(487160680271869523704 n^{4}+75939762859819397480629 n^{3}+4433908294244093852439911 n^{2}+114914729930917758228021821 n +1115351114560041796054539675\right) a \! \left(n +40\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{16 \left(1087032340606801880006 n^{4}+179742915788599467010231 n^{3}+11140154097191743469175685 n^{2}+306727000969785671072618906 n +3165580180294093704426004632\right) a \! \left(n +41\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{16 \left(66172719946724948876 n^{4}+8251045545331000367864 n^{3}+339222445995240981962119 n^{2}+4440181676973269998460224 n -6507171369006882637373019\right) a \! \left(n +42\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{8 \left(2002688348395798881632 n^{4}+343822326547085193603210 n^{3}+22123958775552366350703757 n^{2}+632392718762642830943959503 n +6775186658195412493757919084\right) a \! \left(n +43\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{128 \left(771154980592684759 n^{4}+47317714621477946227 n^{3}+126747450463682959895 n^{2}-36842133377188180859791 n -566434676514864851217978\right) a \! \left(n +30\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{128 \left(16307829372142250175 n^{4}+1997121185423984943160 n^{3}+91726158462409124583039 n^{2}+1872531651848514935584916 n +14335250852349452155146612\right) a \! \left(n +31\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{128 \left(13573465727183953168 n^{4}+1839017978828162808041 n^{3}+93121765005515924859332 n^{2}+2089300752242408138360953 n +17529118248654045169473834\right) a \! \left(n +32\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{32 \left(80457702229811026615 n^{4}+10277772825719334269218 n^{3}+491918068643663214463973 n^{2}+10454395200976119987063842 n +83232084995130334749605352\right) a \! \left(n +33\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{64 \left(87013451631030088130 n^{4}+12143741431478053945842 n^{3}+635062760943348615783979 n^{2}+14749154343797339089180428 n +128357264428661347937504877\right) a \! \left(n +34\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{128 \left(1099833993716753186 n^{4}+83077262212726674628 n^{3}+641244386258131766925 n^{2}-71790885308478075658406 n -1386706264348451869441995\right) a \! \left(n +35\right)}{103278004155 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{1024 \left(3461649296908784 n^{4}+276353284631358181 n^{3}+8027563352940258847 n^{2}+99232349069745704258 n +429543836177211648080\right) a \! \left(n +23\right)}{103278004155 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{512 \left(219126749500927389 n^{4}+19907274040613210774 n^{3}+675957741625954643373 n^{2}+10163209980901351092202 n +57062305972421780632974\right) a \! \left(n +24\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{256 \left(81298607052803859 n^{4}+7512318637077957366 n^{3}+257767183381847016543 n^{2}+3884055638750673822932 n +21620764288507998196576\right) a \! \left(n +25\right)}{103278004155 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{1024 \left(88658948161926150 n^{4}+8867259702346014839 n^{3}+332261550760141941814 n^{2}+5527654845625414804489 n +34445785863419995102356\right) a \! \left(n +26\right)}{103278004155 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{128 \left(2348702265479346191 n^{4}+246411029458317820950 n^{3}+9678125816124196725289 n^{2}+168624130374818001610026 n +1099419741660121071495792\right) a \! \left(n +27\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{256 \left(571109290958408656 n^{4}+60999226094847353133 n^{3}+2440537248560100332598 n^{2}+43346611389370933986709 n +288339009487040262320308\right) a \! \left(n +28\right)}{103278004155 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{256 \left(3835774289530143674 n^{4}+439648224913179281335 n^{3}+18894175497249876424594 n^{2}+360808186451973168996041 n +2583048644083419488245224\right) a \! \left(n +29\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{2048 \left(57069765592389 n^{4}+3916831366396870 n^{3}+102623781442081491 n^{2}+1202576442403416098 n +5276592240925562160\right) a \! \left(n +15\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{4096 \left(130329435989544 n^{4}+7858123245362531 n^{3}+175608347593814889 n^{2}+1722273987379459135 n +6247286428730808537\right) a \! \left(n +16\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{4096 \left(25232851528355 n^{4}+485770374922187 n^{3}-25983028433706662 n^{2}-803975789608392284 n -5875812638897956344\right) a \! \left(n +17\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{4096 \left(806090483698005 n^{4}+54151726525611149 n^{3}+1348020835703540256 n^{2}+14711418089093134741 n +59245818279129513489\right) a \! \left(n +18\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{2048 \left(21324407406135 n^{4}-5562731451858208 n^{3}-392259503772233646 n^{2}-8281688142921335723 n -56667853560439477452\right) a \! \left(n +19\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{1024 \left(12439070370349157 n^{4}+916779800277572623 n^{3}+25040701148372707855 n^{2}+299714772838852050965 n +1322084858763008196024\right) a \! \left(n +20\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{512 \left(3798918535732841 n^{4}+323225411597715250 n^{3}+10373581677560205811 n^{2}+148856432627917334114 n +805877838230515386384\right) a \! \left(n +21\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{1024 \left(38348132708240155 n^{4}+3123535115178892521 n^{3}+94618210193625036983 n^{2}+1261436719820532296175 n +6232636594568712007614\right) a \! \left(n +22\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{32768 \left(11330511841 n^{4}+432729865310 n^{3}+6040465672061 n^{2}+36679614030100 n +81982780990320\right) a \! \left(n +8\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{8192 \left(205755623261 n^{4}+8180488898182 n^{3}+121732000974343 n^{2}+801852172051550 n +1969591000374984\right) a \! \left(n +9\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{16384 \left(77990113978 n^{4}+2776039577486 n^{3}+36757259562089 n^{2}+214682896280728 n +467190996030015\right) a \! \left(n +10\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{32768 \left(414720153889 n^{4}+19716077899969 n^{3}+353236764658718 n^{2}+2817635999686979 n +8420961741703611\right) a \! \left(n +11\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{8192 \left(698890376449 n^{4}+32859037828846 n^{3}+579579939132645 n^{2}+4548480368005470 n +13409962010011054\right) a \! \left(n +12\right)}{103278004155 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{4096 \left(16133127163687 n^{4}+890275413543558 n^{3}+18636060317105117 n^{2}+174676656061811934 n +616323868607483148\right) a \! \left(n +13\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{4096 \left(19597063803219 n^{4}+1057058190141359 n^{3}+21242472421593489 n^{2}+188632660001572741 n +625122551351890752\right) a \! \left(n +14\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{229376 \left(349743 n^{4}+7127342 n^{3}+45011793 n^{2}+112581922 n +96551280\right) a \! \left(n +3\right)}{103278004155 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{458752 \left(2950330 n^{4}+56070545 n^{3}+387848414 n^{2}+1156833409 n +1254660690\right) a \! \left(n +4\right)}{61966802493 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{458752 \left(9723808 n^{4}+128550659 n^{3}+402209030 n^{2}-660348797 n -3337789620\right) a \! \left(n +5\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{229376 \left(379420953 n^{4}+10725608453 n^{3}+111171519813 n^{2}+501545170243 n +831939827250\right) a \! \left(n +6\right)}{309834012465 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}-\frac{65536 \left(723381412 n^{4}+21512094545 n^{3}+238601181408 n^{2}+1169536583638 n +2136999987122\right) a \! \left(n +7\right)}{103278004155 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}+\frac{12845056 \left(n +1\right) \left(4006 n^{3}+35287 n^{2}+102094 n +95277\right) a \! \left(n +2\right)}{61966802493 \left(n +75\right) \left(n +78\right) \left(n +77\right) \left(n +76\right)}, \quad n \geq 77\)

This specification was found using the strategy pack "Row Placements Tracked Fusion Req Corrob" and has 81 rules.

Found on June 15, 2021.

Finding the specification took 139 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_{8}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)+F_{71}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{58}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{3}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{8}\! \left(x \right)}\\ F_{21}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{22}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x \right)+F_{31}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{18}\! \left(x \right) F_{3}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{33}\! \left(x \right) &= -F_{27}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{0}\! \left(x \right) F_{18}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{43}\! \left(x \right) &= -F_{46}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{18}\! \left(x \right) F_{48}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{17}\! \left(x \right) F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{45}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{18}\! \left(x \right) F_{55}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{36}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{0}\! \left(x \right) F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{0} \left(x \right)^{2} F_{3} \left(x \right)^{2} F_{8}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{0} \left(x \right)^{2} F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Row Placements Tracked Fusion" and has 104 rules.

Found on June 15, 2021.

Finding the specification took 84 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_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{17}\! \left(x \right) &= 0\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{6}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x , 1\right)\\ F_{30}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{31}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= -\frac{y \left(F_{32}\! \left(x , 1\right)-F_{32}\! \left(x , y\right)\right)}{-1+y}\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{33}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{37}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= y x\\ F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{42}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{46}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{47}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{50}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{36}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{52}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{57}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{59}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{61}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{63}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{60}\! \left(x , 1\right)\\ F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{2}\! \left(x \right) F_{36}\! \left(x , y\right) F_{69}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)+F_{43}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{102}\! \left(x , y\right)+F_{17}\! \left(x \right)+F_{71}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{72}\! \left(x , y\right)\\ F_{72}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{73}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{73}\! \left(x , y\right) &= F_{74}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{75}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{76}\! \left(x , y\right)+F_{82}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{77}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{77}\! \left(x , y\right) &= F_{78}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\ F_{78}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= F_{80}\! \left(x , y\right)\\ F_{80}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{81}\! \left(x , y\right) &= F_{79}\! \left(x , y\right)\\ F_{82}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{83}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= F_{79}\! \left(x , y\right)+F_{8}\! \left(x \right)\\ F_{85}\! \left(x , y\right) &= F_{86}\! \left(x , y\right)+F_{88}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= F_{87}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= y \,x^{2}\\ F_{88}\! \left(x , y\right) &= F_{89}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{8}\! \left(x \right) F_{90}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{93}\! \left(x , y\right)+F_{95}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\ F_{93}\! \left(x , y\right) &= F_{94}\! \left(x , y\right)\\ F_{94}\! \left(x , y\right) &= F_{8} \left(x \right)^{2} F_{73}\! \left(x , y\right)\\ F_{95}\! \left(x , y\right) &= F_{96}\! \left(x , y\right)\\ F_{96}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{8}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{98}\! \left(x , y\right) &= F_{99}\! \left(x , y\right)\\ F_{99}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{43}\! \left(x , y\right) F_{58}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)\\ F_{101}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{36}\! \left(x , y\right) F_{69}\! \left(x , y\right)\\ F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)\\ F_{103}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)^{2} F_{43}\! \left(x , y\right) F_{58}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 27 rules.

Found on January 22, 2022.

Finding the specification took 49 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_{7}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\ F_{4}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{5}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{7}\! \left(x \right)\\ F_{6}\! \left(x , y\right) &= -\frac{-y F_{4}\! \left(x , y\right)+F_{4}\! \left(x , 1\right)}{-1+y}\\ F_{7}\! \left(x \right) &= x\\ F_{8}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , 1, y\right)\\ F_{10}\! \left(x , y , z\right) &= F_{11}\! \left(x , y z , z\right)\\ F_{11}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)+F_{22}\! \left(x , y , z\right)\\ F_{12}\! \left(x , y , z\right) &= F_{13}\! \left(x , y , z\right)\\ F_{13}\! \left(x , y , z\right) &= F_{14}\! \left(x , y , z\right) F_{7}\! \left(x \right)\\ F_{14}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right)+F_{6}\! \left(x , z\right)\\ F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , z\right) F_{17}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{11}\! \left(x , 1, y\right)\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x , y , z\right) &= F_{19}\! \left(x , y , z\right)\\ F_{19}\! \left(x , y , z\right) &= F_{17}\! \left(x , y\right) F_{20}\! \left(x , y\right) F_{4}\! \left(x , z\right)\\ F_{20}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)\\ F_{22}\! \left(x , y , z\right) &= F_{23}\! \left(x , y , z\right)\\ F_{23}\! \left(x , y , z\right) &= F_{17}\! \left(x , z\right) F_{24}\! \left(x , y , z\right)\\ F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , y , z\right)+F_{9}\! \left(x , z\right)\\ F_{25}\! \left(x , y , z\right) &= F_{26}\! \left(x , y , z\right)\\ F_{26}\! \left(x , y , z\right) &= F_{17}\! \left(x , y\right) F_{21}\! \left(x , z\right) F_{4}\! \left(x , z\right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Row And Col Placements Tracked Fusion Req Corrob" and has 178 rules.

Found on June 15, 2021.

Finding the specification took 218 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_{13}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{13}\! \left(x \right) F_{4}\! \left(x \right)}\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{0}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{35}\! \left(x \right) &= 0\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{2} \left(x \right)^{2}\\ F_{45}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{11}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{13}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{2}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{30}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= \frac{F_{62}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{0}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= -F_{102}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= -F_{101}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= -F_{96}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= \frac{F_{68}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{13}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{2}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{13}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{80}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{13}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{13}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{13}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{2}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= -F_{74}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= \frac{F_{94}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{94}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{2} \left(x \right)^{2} F_{0}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{2}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{113}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{2}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{2}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{2} \left(x \right)^{2} F_{13}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{153}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{150}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{59}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{122}\! \left(x \right) &= -F_{44}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{123}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= \frac{F_{125}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)\\ F_{126}\! \left(x \right) &= -F_{148}\! \left(x \right)-F_{35}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{147}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{145}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{133}\! \left(x \right) &= -F_{144}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{134}\! \left(x \right) &= -F_{13}\! \left(x \right)+F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= \frac{F_{136}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{142}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{141}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{2}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{146}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{152}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{59}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{121}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{176}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{13}\! \left(x \right) F_{156}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{173}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{159}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{39}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{161}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{2}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{161}\! \left(x \right) &= -F_{172}\! \left(x \right)+F_{162}\! \left(x \right)\\ F_{162}\! \left(x \right) &= -F_{171}\! \left(x \right)+F_{163}\! \left(x \right)\\ F_{163}\! \left(x \right) &= \frac{F_{164}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)\\ F_{165}\! \left(x \right) &= -F_{170}\! \left(x \right)-F_{35}\! \left(x \right)+F_{166}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{13}\! \left(x \right) F_{168}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{0}\! \left(x \right) F_{127}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{101}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{2} \left(x \right)^{2} F_{13}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{175}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{33}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{159}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right) F_{55}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Row And Col Placements Tracked Fusion" and has 178 rules.

Found on June 15, 2021.

Finding the specification took 135 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_{13}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{13}\! \left(x \right) F_{4}\! \left(x \right)}\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{0}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{35}\! \left(x \right) &= 0\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{2} \left(x \right)^{2}\\ F_{45}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{11}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{13}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{2}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{30}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= \frac{F_{62}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{0}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= -F_{102}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= -F_{101}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= -F_{96}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= \frac{F_{68}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{13}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{2}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{13}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{80}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{13}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{13}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{13}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{2}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= -F_{74}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= \frac{F_{94}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{94}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{2} \left(x \right)^{2} F_{0}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{2}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{113}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{2}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{2}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{2} \left(x \right)^{2} F_{13}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{153}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{150}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{59}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{122}\! \left(x \right) &= -F_{44}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{123}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= \frac{F_{125}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)\\ F_{126}\! \left(x \right) &= -F_{148}\! \left(x \right)-F_{35}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{147}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{145}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{133}\! \left(x \right) &= -F_{144}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{134}\! \left(x \right) &= -F_{13}\! \left(x \right)+F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= \frac{F_{136}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{142}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{141}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{2}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{146}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{152}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{59}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{121}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{176}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{13}\! \left(x \right) F_{156}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{173}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{159}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{39}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{161}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{2}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{161}\! \left(x \right) &= -F_{172}\! \left(x \right)+F_{162}\! \left(x \right)\\ F_{162}\! \left(x \right) &= -F_{171}\! \left(x \right)+F_{163}\! \left(x \right)\\ F_{163}\! \left(x \right) &= \frac{F_{164}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)\\ F_{165}\! \left(x \right) &= -F_{170}\! \left(x \right)-F_{35}\! \left(x \right)+F_{166}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{13}\! \left(x \right) F_{168}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{0}\! \left(x \right) F_{127}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{101}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{2} \left(x \right)^{2} F_{13}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{175}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{33}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{159}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right) F_{55}\! \left(x \right)\\ \end{align*}\)