Av(13254, 13524, 13542, 15324, 15342, 31254, 31524, 31542)
Counting Sequence
1, 1, 2, 6, 24, 112, 562, 2931, 15690, 85753, 476968, 2692904, 15398650, 89014193, 519366248, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x -1\right) x F \left(x
\right)^{6}+\left(-16 x^{2}-2 x +1\right) F \left(x
\right)^{5}+\left(16 x^{2}+26 x -3\right) F \left(x
\right)^{4}+\left(-48 x -5\right) F \left(x
\right)^{3}+\left(24 x +25\right) F \left(x
\right)^{2}-27 F \! \left(x \right)+9 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 112\)
\(\displaystyle a \! \left(6\right) = 562\)
\(\displaystyle a \! \left(7\right) = 2931\)
\(\displaystyle a \! \left(8\right) = 15690\)
\(\displaystyle a \! \left(9\right) = 85753\)
\(\displaystyle a \! \left(10\right) = 476968\)
\(\displaystyle a \! \left(11\right) = 2692904\)
\(\displaystyle a \! \left(12\right) = 15398650\)
\(\displaystyle a \! \left(13\right) = 89014193\)
\(\displaystyle a \! \left(14\right) = 519366248\)
\(\displaystyle a \! \left(15\right) = 3054715077\)
\(\displaystyle a \! \left(16\right) = 18092383108\)
\(\displaystyle a \! \left(17\right) = 107814079392\)
\(\displaystyle a \! \left(18\right) = 645952458506\)
\(\displaystyle a \! \left(19\right) = 3888772631544\)
\(\displaystyle a \! \left(20\right) = 23512210755034\)
\(\displaystyle a \! \left(21\right) = 142711110962299\)
\(\displaystyle a \! \left(22\right) = 869253312772618\)
\(\displaystyle a \! \left(23\right) = 5311553879372443\)
\(\displaystyle a \! \left(24\right) = 32550957168642214\)
\(\displaystyle a \! \left(25\right) = 200017510844811693\)
\(\displaystyle a \! \left(26\right) = 1232088802805538738\)
\(\displaystyle a \! \left(27\right) = 7606834873896719932\)
\(\displaystyle a \! \left(28\right) = 47063175955743984188\)
\(\displaystyle a \! \left(29\right) = 291748588176606541158\)
\(\displaystyle a \! \left(30\right) = 1811875252278745495856\)
\(\displaystyle a \! \left(n +31\right) = \frac{24046319299461120 \left(2 n +3\right) \left(n +2\right) \left(n +1\right) \left(29731 n^{2}+152613 n +197700\right) a \! \left(n +1\right)}{12131 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{25048249270272 \left(n +2\right) \left(245617983 n^{4}+3015005864 n^{3}+13862355557 n^{2}+28254155316 n +21510584640\right) a \! \left(n +2\right)}{12131 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(582048882082744 n^{4}+65572088971265424 n^{3}+2757191584395862205 n^{2}+51313295333754781257 n +356797097475552592182\right) a \! \left(n +27\right)}{1552768000 \left(n +30\right) \left(n +29\right) \left(n +32\right) \left(n +31\right)}+\frac{\left(1831118728090 n^{3}+150062500811607 n^{2}+4097910052131881 n +37290005274048714\right) a \! \left(n +28\right)}{38819200 \left(n +30\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(49836647801 n^{2}+2742764145823 n +37741815985512\right) a \! \left(n +29\right)}{19409600 \left(n +32\right) \left(n +31\right)}-\frac{5771116631870668800 n \left(2 n +3\right) \left(2 n +1\right) \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{1733 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{3 \left(12309991 n +343302762\right) a \! \left(n +30\right)}{485240 \left(n +32\right)}+\frac{\left(754247901906419054347613 n^{5}+70587046077894686264553120 n^{4}+2642103739018935803971454525 n^{3}+49443529417029329299231644885 n^{2}+462609403017216405176197221917 n +1731272118193106292671139737790\right) a \! \left(n +19\right)}{4852400000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(103620917227196614344073 n^{5}+10163891638838734388798474 n^{4}+398679520516378816796541494 n^{3}+7817308606264556793546655798 n^{2}+76624567664124226713153139851 n +300368346752959759625655370446\right) a \! \left(n +20\right)}{3881920000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{\left(30320472556834093590629 n^{5}+3108628569364730737700120 n^{4}+127432934278034486649042154 n^{3}+2610893226339518398107687727 n^{2}+26735944683417111094910116638 n +109470009106284226168353262488\right) a \! \left(n +21\right)}{7763840000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(7431442793944725456023 n^{5}+794211367367501702654738 n^{4}+33930551719692965725531141 n^{3}+724347632670704134594676458 n^{2}+7726867122771289236543665472 n +32949480455463356099550289752\right) a \! \left(n +22\right)}{15527680000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{\left(1485422901254642746513 n^{5}+164972150191051379623516 n^{4}+7322155742382199014773499 n^{3}+162343773472387509093431468 n^{2}+1798001147615648690587142204 n +7957558945846738047355428000\right) a \! \left(n +23\right)}{31055360000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(57686303184665061819 n^{5}+6629317627502869425188 n^{4}+304308552719489172046341 n^{3}+6974111892703138293657028 n^{2}+79792029661747033561497096 n +364567551107274100493780400\right) a \! \left(n +24\right)}{15527680000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{3 \left(2073574254782714349 n^{5}+244243024274772780024 n^{4}+11475478456513729442745 n^{3}+268759134843397196154002 n^{2}+3136659495251729791229952 n +14588696197463574550376320\right) a \! \left(n +25\right)}{31055360000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(34594201652788176 n^{5}+3897347598423273217 n^{4}+172062607359326354207 n^{3}+3694072015268268110072 n^{2}+38103626659042879586116 n +147706398512481747246960\right) a \! \left(n +26\right)}{7763840000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{16 \left(302271453752238665505889 n^{5}+21318384897896650271914030 n^{4}+601734343017711464554305870 n^{3}+8497212175011809279598168245 n^{2}+60032389325312885084400999686 n +169760661425372374325270936535\right) a \! \left(n +14\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{4 \left(82206705529424544488412 n^{5}+6181261648260323995695104 n^{4}+185987945805716541369184944 n^{3}+2799351310797833739496853161 n^{2}+21077110095694146912477807519 n +63511267667607892954723503141\right) a \! \left(n +15\right)}{7581875 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{3 \left(163865910772739526792856 n^{5}+13082186362475631923067240 n^{4}+417883126774076935876590380 n^{3}+6676336163247255721825745955 n^{2}+53351397228526361022995934469 n +170601277467845813188371121165\right) a \! \left(n +16\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{\left(1031792380605269851936701 n^{5}+87136664087644880595789380 n^{4}+2943997072946191794453347245 n^{3}+49742350122059412838445850520 n^{2}+420321821656325679381535023614 n +1421045696701070689991615754990\right) a \! \left(n +17\right)}{303275000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(189271344348479921812495 n^{5}+16852280921012119587380218 n^{4}+600211817419516323033119126 n^{3}+10689203102126549360708869775 n^{2}+95190524652962517046802834562 n +339118988679000956343030599106\right) a \! \left(n +18\right)}{242620000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{24576 \left(6215534204524538743999 n^{5}+292349202911276382477740 n^{4}+5506746199162866169756355 n^{3}+51924707119375239835729060 n^{2}+245097446400200432845645746 n +463311533260691901545666760\right) a \! \left(n +9\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{6144 \left(3258891576657996993409 n^{5}+168623017896235826726998 n^{4}+3493673675960023118287214 n^{3}+36231683603489053341330053 n^{2}+188080024547162094820069686 n +390964452567407356966590456\right) a \! \left(n +10\right)}{7581875 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{1536 \left(37197314893147533118471 n^{5}+2099759144151741547491760 n^{4}+47455941502919607345810015 n^{3}+536785194335902456706652815 n^{2}+3038863162669705616053042199 n +6888434473642515765576476880\right) a \! \left(n +11\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{128 \left(44598202411812121403345 n^{5}+2727257059029128584261226 n^{4}+66763831471309189643424250 n^{3}+817881215716812434270270482 n^{2}+5014037199311991222239893707 n +12306508320187023110601644268\right) a \! \left(n +12\right)}{7581875 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{64 \left(195697684894799856877214 n^{5}+12885879941160694729723300 n^{4}+339619086178799429180369415 n^{3}+4478655761441657753056887755 n^{2}+29552672240140927086369605056 n +78062605640760941432672328645\right) a \! \left(n +13\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{452984832 \left(15081597096842221 n^{5}+426205613818745660 n^{4}+4822808178398459645 n^{3}+27309110730549610330 n^{2}+77363734805567987424 n +87693322384001985120\right) a \! \left(n +5\right)}{1516375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{37748736 \left(1147201747529839911 n^{5}+37790669318240309510 n^{4}+498590576716976842020 n^{3}+3292919668178114679535 n^{2}+10885222280105246099724 n +14405712166869868946880\right) a \! \left(n +6\right)}{7581875 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{786432 \left(289083820321920588221 n^{5}+10878920513263202712440 n^{4}+163976406928881024367645 n^{3}+1237371884981580536509990 n^{2}+4674256272205046568010764 n +7070818524721208324412480\right) a \! \left(n +7\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{196608 \left(1024673133031818455389 n^{5}+43375422632398155634290 n^{4}+735382963546424009295820 n^{3}+6241673276813163587555505 n^{2}+26521273425283465301096686 n +45129913425906071175016920\right) a \! \left(n +8\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{2087354105856 \left(8213252877 n^{5}+155450189108 n^{4}+1175156838675 n^{3}+4432149941500 n^{2}+8333403760788 n +6244445225952\right) a \! \left(n +3\right)}{12131 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{86973087744 \left(1998626305201 n^{5}+47144168570257 n^{4}+444972277174067 n^{3}+2099833463101091 n^{2}+4952242152500064 n +4667539713463620\right) a \! \left(n +4\right)}{60655 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}, \quad n \geq 31\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 112\)
\(\displaystyle a \! \left(6\right) = 562\)
\(\displaystyle a \! \left(7\right) = 2931\)
\(\displaystyle a \! \left(8\right) = 15690\)
\(\displaystyle a \! \left(9\right) = 85753\)
\(\displaystyle a \! \left(10\right) = 476968\)
\(\displaystyle a \! \left(11\right) = 2692904\)
\(\displaystyle a \! \left(12\right) = 15398650\)
\(\displaystyle a \! \left(13\right) = 89014193\)
\(\displaystyle a \! \left(14\right) = 519366248\)
\(\displaystyle a \! \left(15\right) = 3054715077\)
\(\displaystyle a \! \left(16\right) = 18092383108\)
\(\displaystyle a \! \left(17\right) = 107814079392\)
\(\displaystyle a \! \left(18\right) = 645952458506\)
\(\displaystyle a \! \left(19\right) = 3888772631544\)
\(\displaystyle a \! \left(20\right) = 23512210755034\)
\(\displaystyle a \! \left(21\right) = 142711110962299\)
\(\displaystyle a \! \left(22\right) = 869253312772618\)
\(\displaystyle a \! \left(23\right) = 5311553879372443\)
\(\displaystyle a \! \left(24\right) = 32550957168642214\)
\(\displaystyle a \! \left(25\right) = 200017510844811693\)
\(\displaystyle a \! \left(26\right) = 1232088802805538738\)
\(\displaystyle a \! \left(27\right) = 7606834873896719932\)
\(\displaystyle a \! \left(28\right) = 47063175955743984188\)
\(\displaystyle a \! \left(29\right) = 291748588176606541158\)
\(\displaystyle a \! \left(30\right) = 1811875252278745495856\)
\(\displaystyle a \! \left(n +31\right) = \frac{24046319299461120 \left(2 n +3\right) \left(n +2\right) \left(n +1\right) \left(29731 n^{2}+152613 n +197700\right) a \! \left(n +1\right)}{12131 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{25048249270272 \left(n +2\right) \left(245617983 n^{4}+3015005864 n^{3}+13862355557 n^{2}+28254155316 n +21510584640\right) a \! \left(n +2\right)}{12131 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(582048882082744 n^{4}+65572088971265424 n^{3}+2757191584395862205 n^{2}+51313295333754781257 n +356797097475552592182\right) a \! \left(n +27\right)}{1552768000 \left(n +30\right) \left(n +29\right) \left(n +32\right) \left(n +31\right)}+\frac{\left(1831118728090 n^{3}+150062500811607 n^{2}+4097910052131881 n +37290005274048714\right) a \! \left(n +28\right)}{38819200 \left(n +30\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(49836647801 n^{2}+2742764145823 n +37741815985512\right) a \! \left(n +29\right)}{19409600 \left(n +32\right) \left(n +31\right)}-\frac{5771116631870668800 n \left(2 n +3\right) \left(2 n +1\right) \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{1733 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{3 \left(12309991 n +343302762\right) a \! \left(n +30\right)}{485240 \left(n +32\right)}+\frac{\left(754247901906419054347613 n^{5}+70587046077894686264553120 n^{4}+2642103739018935803971454525 n^{3}+49443529417029329299231644885 n^{2}+462609403017216405176197221917 n +1731272118193106292671139737790\right) a \! \left(n +19\right)}{4852400000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(103620917227196614344073 n^{5}+10163891638838734388798474 n^{4}+398679520516378816796541494 n^{3}+7817308606264556793546655798 n^{2}+76624567664124226713153139851 n +300368346752959759625655370446\right) a \! \left(n +20\right)}{3881920000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{\left(30320472556834093590629 n^{5}+3108628569364730737700120 n^{4}+127432934278034486649042154 n^{3}+2610893226339518398107687727 n^{2}+26735944683417111094910116638 n +109470009106284226168353262488\right) a \! \left(n +21\right)}{7763840000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(7431442793944725456023 n^{5}+794211367367501702654738 n^{4}+33930551719692965725531141 n^{3}+724347632670704134594676458 n^{2}+7726867122771289236543665472 n +32949480455463356099550289752\right) a \! \left(n +22\right)}{15527680000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{\left(1485422901254642746513 n^{5}+164972150191051379623516 n^{4}+7322155742382199014773499 n^{3}+162343773472387509093431468 n^{2}+1798001147615648690587142204 n +7957558945846738047355428000\right) a \! \left(n +23\right)}{31055360000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(57686303184665061819 n^{5}+6629317627502869425188 n^{4}+304308552719489172046341 n^{3}+6974111892703138293657028 n^{2}+79792029661747033561497096 n +364567551107274100493780400\right) a \! \left(n +24\right)}{15527680000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{3 \left(2073574254782714349 n^{5}+244243024274772780024 n^{4}+11475478456513729442745 n^{3}+268759134843397196154002 n^{2}+3136659495251729791229952 n +14588696197463574550376320\right) a \! \left(n +25\right)}{31055360000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(34594201652788176 n^{5}+3897347598423273217 n^{4}+172062607359326354207 n^{3}+3694072015268268110072 n^{2}+38103626659042879586116 n +147706398512481747246960\right) a \! \left(n +26\right)}{7763840000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{16 \left(302271453752238665505889 n^{5}+21318384897896650271914030 n^{4}+601734343017711464554305870 n^{3}+8497212175011809279598168245 n^{2}+60032389325312885084400999686 n +169760661425372374325270936535\right) a \! \left(n +14\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{4 \left(82206705529424544488412 n^{5}+6181261648260323995695104 n^{4}+185987945805716541369184944 n^{3}+2799351310797833739496853161 n^{2}+21077110095694146912477807519 n +63511267667607892954723503141\right) a \! \left(n +15\right)}{7581875 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{3 \left(163865910772739526792856 n^{5}+13082186362475631923067240 n^{4}+417883126774076935876590380 n^{3}+6676336163247255721825745955 n^{2}+53351397228526361022995934469 n +170601277467845813188371121165\right) a \! \left(n +16\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{\left(1031792380605269851936701 n^{5}+87136664087644880595789380 n^{4}+2943997072946191794453347245 n^{3}+49742350122059412838445850520 n^{2}+420321821656325679381535023614 n +1421045696701070689991615754990\right) a \! \left(n +17\right)}{303275000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{\left(189271344348479921812495 n^{5}+16852280921012119587380218 n^{4}+600211817419516323033119126 n^{3}+10689203102126549360708869775 n^{2}+95190524652962517046802834562 n +339118988679000956343030599106\right) a \! \left(n +18\right)}{242620000 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{24576 \left(6215534204524538743999 n^{5}+292349202911276382477740 n^{4}+5506746199162866169756355 n^{3}+51924707119375239835729060 n^{2}+245097446400200432845645746 n +463311533260691901545666760\right) a \! \left(n +9\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{6144 \left(3258891576657996993409 n^{5}+168623017896235826726998 n^{4}+3493673675960023118287214 n^{3}+36231683603489053341330053 n^{2}+188080024547162094820069686 n +390964452567407356966590456\right) a \! \left(n +10\right)}{7581875 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{1536 \left(37197314893147533118471 n^{5}+2099759144151741547491760 n^{4}+47455941502919607345810015 n^{3}+536785194335902456706652815 n^{2}+3038863162669705616053042199 n +6888434473642515765576476880\right) a \! \left(n +11\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{128 \left(44598202411812121403345 n^{5}+2727257059029128584261226 n^{4}+66763831471309189643424250 n^{3}+817881215716812434270270482 n^{2}+5014037199311991222239893707 n +12306508320187023110601644268\right) a \! \left(n +12\right)}{7581875 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{64 \left(195697684894799856877214 n^{5}+12885879941160694729723300 n^{4}+339619086178799429180369415 n^{3}+4478655761441657753056887755 n^{2}+29552672240140927086369605056 n +78062605640760941432672328645\right) a \! \left(n +13\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{452984832 \left(15081597096842221 n^{5}+426205613818745660 n^{4}+4822808178398459645 n^{3}+27309110730549610330 n^{2}+77363734805567987424 n +87693322384001985120\right) a \! \left(n +5\right)}{1516375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{37748736 \left(1147201747529839911 n^{5}+37790669318240309510 n^{4}+498590576716976842020 n^{3}+3292919668178114679535 n^{2}+10885222280105246099724 n +14405712166869868946880\right) a \! \left(n +6\right)}{7581875 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{786432 \left(289083820321920588221 n^{5}+10878920513263202712440 n^{4}+163976406928881024367645 n^{3}+1237371884981580536509990 n^{2}+4674256272205046568010764 n +7070818524721208324412480\right) a \! \left(n +7\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{196608 \left(1024673133031818455389 n^{5}+43375422632398155634290 n^{4}+735382963546424009295820 n^{3}+6241673276813163587555505 n^{2}+26521273425283465301096686 n +45129913425906071175016920\right) a \! \left(n +8\right)}{37909375 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}+\frac{2087354105856 \left(8213252877 n^{5}+155450189108 n^{4}+1175156838675 n^{3}+4432149941500 n^{2}+8333403760788 n +6244445225952\right) a \! \left(n +3\right)}{12131 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}-\frac{86973087744 \left(1998626305201 n^{5}+47144168570257 n^{4}+444972277174067 n^{3}+2099833463101091 n^{2}+4952242152500064 n +4667539713463620\right) a \! \left(n +4\right)}{60655 \left(n +30\right) \left(n +29\right) \left(n +28\right) \left(n +32\right) \left(n +31\right)}, \quad n \geq 31\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 196 rules.
Found on January 25, 2022.Finding the specification took 2016 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_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{23}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{195}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{17}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= x\\
F_{24}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{23}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= \frac{F_{28}\! \left(x \right)}{F_{23}\! \left(x \right) F_{34}\! \left(x \right)}\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= -F_{33}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= \frac{F_{31}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= -F_{4}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{0}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{23}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= \frac{F_{38}\! \left(x \right)}{F_{182}\! \left(x \right) F_{23}\! \left(x \right)}\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= -F_{48}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= \frac{F_{41}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{23}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= \frac{F_{44}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= \frac{F_{47}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{47}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{48}\! \left(x \right) &= -F_{49}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{50}\! \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_{99}\! \left(x \right)\\
F_{52}\! \left(x \right) &= -F_{94}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= -F_{56}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= \frac{F_{55}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{55}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{0}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= \frac{F_{60}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{60}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{23}\! \left(x \right) F_{63}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \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_{69}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{17}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{23}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{23}\! \left(x \right) F_{75}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= \frac{F_{79}\! \left(x \right)}{F_{23}\! \left(x \right) F_{34}\! \left(x \right)}\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{23}\! \left(x \right) F_{81}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{23}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{90}\! \left(x \right) &= 0\\
F_{91}\! \left(x \right) &= F_{23}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{23}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{23}\! \left(x \right) F_{81}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{23}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{23}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{23}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{133}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{23}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{114}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{120}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{116}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{125}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{130}\! \left(x \right)+F_{132}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{121}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{151}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{137}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{145}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{143}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{148}\! \left(x \right)+F_{150}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{139}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{155}\! \left(x \right)+F_{157}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{152}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{152}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{166}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{163}\! \left(x \right)+F_{165}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{160}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{164}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{159}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)+F_{174}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)+F_{170}\! \left(x \right)+F_{172}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{169}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{171}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{173}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{167}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{175}\! \left(x \right)+F_{177}\! \left(x \right)+F_{179}\! \left(x \right)+F_{181}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{176}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{174}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{178}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{174}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{180}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{174}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{166}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)+F_{189}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{184}\! \left(x \right)+F_{187}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)+F_{186}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{108}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{188}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{189}\! \left(x \right) &= F_{190}\! \left(x \right)+F_{193}\! \left(x \right)\\
F_{190}\! \left(x \right) &= F_{191}\! \left(x \right)+F_{192}\! \left(x \right)\\
F_{191}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{192}\! \left(x \right) &= F_{134}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{194}\! \left(x \right)\\
F_{194}\! \left(x \right) &= F_{152}\! \left(x \right)\\
F_{195}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
\end{align*}\)