0123_0321_1302_2031
Counting sequence:
1, 1, 2, 6, 20, 60, 173, 503, 1479, 4360, 12839, 37781, 111183, 327252, 963283, 2835454, 8346162, 24566934, 72312996, 212854265, 626539434, 1844227160, 5428507154, 15978883706, 47034060140, 138445392609, 407515886846, 1199528528447, 3530828461603, 10393041380296, 30592058019909, 90048136987756, 265057910449486, 780201548231799, 2296533821003379, 6759878396802419, 19897793588634123, 58569424841054599, 172399895040281684, 507461425318221826, 1493719576371396306, 4396783797775628912, 12941992640508976118, 38094930569869779912, 112132943931742069887, 330064838725397012197, 971550321813886936898, 2859771526896780055055, 8417776210274574366911, 24777838250300950997460, 72933902377769577917364, 214681929162460911950276, 631919165249050432108943, 1860062618996074154601780, 5475119503968444553679515, 16116088392182744955021784, 47437924391676317035159941, 139634172748888298274690271, 411015078110954168815838367, 1209828447498706759922739675, 3561146416098163236939281429, 10482282693143854582316336073, 30854741035717746179157854153, 90821348006945947486702827394, 267333867565124640037251366249, 786900858836204765350114568991, 2316253332497467777147449459000, 6817923045909863877962027639725, 20068648799234767693396205259051, 59072339466876352123703815595245, 173880231050878536263151977518281, 511818814409072294560074815284432, 1506545610158854424300698913689658, 4434537401891733860411628819251479, 13053120885402960425263276316394487, 38422038063374704004146516889660882, 113095789267858408226917439351246769, 332898987009030139009941888751662590, 979892675660681981025428692282986853, 2884327358398974689167776620445830200, 8490056632783360638170010559625663317, 24990596652621051155908009222744776390, 73560159615713898267321132108161161154, 216525325821773924460623858774129413111, 637345228275037344233276293797512144090, 1876034308981101334447668986164107953848, 5522132389693526918653755313731230288381, 16254471457861557723556300102036771962090, 47845256819186712216279988968836633030599, 140833161264481753417317046971036655512968, 414544317040718261526033396611533233488404, 1220216810073803962630127671436172272853944, 3591724701994747128452125759800273161159955, 10572290291705601195091108485433593103211178, 31119679620772887868597073500475408602436430, 91601198319282271313920103127392195008841873, 269629367518536424437676174348350070745025671, 793657694029778451679551487011154562488672964, 2336142168376231152946936677043642668532016278, 6876466103610694200840567431526714321298770021, 20240971082240898390860911709478872223340458517
Generating function in Maple syntax:
(x-1)*(x^3+2*x-1)/(x^8+2*x^7+3*x^6+2*x^5-3*x^3+4*x^2-4*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(x^{3}+2 x -1\right)}{x^{8}+2 x^{7}+3 x^{6}+2 x^{5}-3 x^{3}+4 x^{2}-4 x +1}
Generating function in sympy syntax:
(x - 1)*(x**3 + 2*x - 1)/(x**8 + 2*x**7 + 3*x**6 + 2*x**5 - 3*x**3 + 4*x**2 - 4*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^8+2*x^7+3*x^6+2*x^5-3*x^3+4*x^2-4*x+1)*F(x)-(x-1)*(x^3+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{8}+2 x^{7}+3 x^{6}+2 x^{5}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{3}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
-397775981/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+6)-397775981/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+6)-397775981/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+6)-397775981/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+6)-397775981/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+6)-397775981/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+6)-397775981/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+6)-397775981/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+6)-1288294133/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+5)-1288294133/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+5)-1288294133/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+5)-1288294133/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+5)-1288294133/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+5)-1288294133/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+5)-1288294133/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+5)-1288294133/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+5)-2617452305/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+4)-2617452305/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+4)-2617452305/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+4)-2617452305/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+4)-2617452305/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+4)-2617452305/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+4)-2617452305/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+4)-2617452305/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+4)-3108083525/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+3)-3108083525/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+3)-3108083525/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+3)-3108083525/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+3)-3108083525/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+3)-3108083525/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+3)-3108083525/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+3)-3108083525/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+3)-1604501690/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+2)-1604501690/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+2)-1604501690/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+2)-1604501690/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+2)-1604501690/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+2)-1604501690/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+2)-1604501690/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+2)-1604501690/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+2)+951788788/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+1)+951788788/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+1)+951788788/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+1)+951788788/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+1)+951788788/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+1)+951788788/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+1)+951788788/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+1)+951788788/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+1)+1014002868/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n-1)+1014002868/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n-1)+1014002868/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n-1)+1014002868/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n-1)+1014002868/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n-1)+1014002868/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n-1)+1014002868/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n-1)+1014002868/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n-1)+1798415758/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n)+1798415758/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n)+1798415758/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n)+1798415758/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n)+1798415758/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n)+1798415758/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n)+1798415758/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n)+1798415758/18105441281*RootOf(_Z^8+2*_Z^7+3*_Z^6+2*_Z^5-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n)
Explicit closed form in latex syntax:
-\frac{397775981 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{18105441281}-\frac{397775981 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{18105441281}-\frac{397775981 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{18105441281}-\frac{397775981 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{18105441281}-\frac{397775981 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{18105441281}-\frac{397775981 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{18105441281}-\frac{397775981 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{18105441281}-\frac{397775981 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{18105441281}-\frac{1288294133 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{18105441281}-\frac{1288294133 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{18105441281}-\frac{1288294133 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{18105441281}-\frac{1288294133 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{18105441281}-\frac{1288294133 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{18105441281}-\frac{1288294133 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{18105441281}-\frac{1288294133 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{18105441281}-\frac{1288294133 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{18105441281}-\frac{2617452305 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{18105441281}-\frac{2617452305 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{18105441281}-\frac{2617452305 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{18105441281}-\frac{2617452305 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{18105441281}-\frac{2617452305 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{18105441281}-\frac{2617452305 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{18105441281}-\frac{2617452305 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{18105441281}-\frac{2617452305 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{18105441281}-\frac{3108083525 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{18105441281}-\frac{3108083525 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{18105441281}-\frac{3108083525 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{18105441281}-\frac{3108083525 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{18105441281}-\frac{3108083525 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{18105441281}-\frac{3108083525 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{18105441281}-\frac{3108083525 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{18105441281}-\frac{3108083525 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{18105441281}-\frac{1604501690 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{18105441281}-\frac{1604501690 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{18105441281}-\frac{1604501690 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{18105441281}-\frac{1604501690 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{18105441281}-\frac{1604501690 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{18105441281}-\frac{1604501690 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{18105441281}-\frac{1604501690 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{18105441281}-\frac{1604501690 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{18105441281}+\frac{951788788 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{18105441281}+\frac{951788788 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{18105441281}+\frac{951788788 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{18105441281}+\frac{951788788 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{18105441281}+\frac{951788788 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{18105441281}+\frac{951788788 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{18105441281}+\frac{951788788 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{18105441281}+\frac{951788788 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{18105441281}+\frac{1014002868 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{18105441281}+\frac{1014002868 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{18105441281}+\frac{1014002868 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{18105441281}+\frac{1014002868 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{18105441281}+\frac{1014002868 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{18105441281}+\frac{1014002868 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{18105441281}+\frac{1014002868 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{18105441281}+\frac{1014002868 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{18105441281}+\frac{1798415758 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{18105441281}+\frac{1798415758 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{18105441281}+\frac{1798415758 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{18105441281}+\frac{1798415758 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{18105441281}+\frac{1798415758 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{18105441281}+\frac{1798415758 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{18105441281}+\frac{1798415758 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{18105441281}+\frac{1798415758 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{18105441281}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 60
a(6) = 173
a(7) = 503
a(n+3) = -1/2*a(n)-a(n+1)-3/2*a(n+2)+3/2*a(n+5)-2*a(n+6)+2*a(n+7)-1/2*a(n+8), n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 60
a \! \left(6\right) = 173
a \! \left(7\right) = 503
a \! \left(n +3\right) = -\frac{a \! \left(n \right)}{2}-a \! \left(n +1\right)-\frac{3 a \! \left(n +2\right)}{2}+\frac{3 a \! \left(n +5\right)}{2}-2 a \! \left(n +6\right)+2 a \! \left(n +7\right)-\frac{a \! \left(n +8\right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15115/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[23,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[15,x]
F[23,x] = F[2,x]+F[24,x]
F[24,x] = F[18,x]+F[25,x]+F[60,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[33,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[11,x]
F[32,x] = F[19,x]
F[33,x] = F[24,x]+F[34,x]
F[34,x] = 2*F[18,x]+F[35,x]+F[39,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[28,x]
F[38,x] = F[34,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[57,x]
F[41,x] = F[42,x]
F[42,x] = F[18,x]+F[43,x]+F[53,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = F[11,x]+F[46,x]
F[46,x] = F[29,x]
F[47,x] = F[42,x]+F[48,x]
F[48,x] = 2*F[18,x]+F[39,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[46,x]
F[52,x] = F[48,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[2,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[2,x]*F[4,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[42,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[2,x]+F[42,x]
F[63,x] = F[64,x]+F[66,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[54,x]
F[66,x] = 2*F[18,x]+F[58,x]+F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[64,x]
System of equations in latex syntax:
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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{23}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \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_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)
F_{23}\! \left(x \right) = F_{2}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)+F_{60}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{33}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right)
F_{32}\! \left(x \right) = F_{19}\! \left(x \right)
F_{33}\! \left(x \right) = F_{24}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{28}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{57}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{43}\! \left(x \right)+F_{53}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{29}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{46}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{2}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{42}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{2}\! \left(x \right)+F_{42}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{58}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{64}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_23(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_15(x))
Eq(F_23(x), F_2(x) + F_24(x))
Eq(F_24(x), F_18(x) + F_25(x) + F_60(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_33(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_11(x))
Eq(F_32(x), F_19(x))
Eq(F_33(x), F_24(x) + F_34(x))
Eq(F_34(x), 2*F_18(x) + F_35(x) + F_39(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_28(x))
Eq(F_38(x), F_34(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_57(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_18(x) + F_43(x) + F_53(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_11(x) + F_46(x))
Eq(F_46(x), F_29(x))
Eq(F_47(x), F_42(x) + F_48(x))
Eq(F_48(x), 2*F_18(x) + F_39(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_46(x))
Eq(F_52(x), F_48(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_2(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_2(x)*F_4(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_42(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_2(x) + F_42(x))
Eq(F_63(x), F_64(x) + F_66(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_54(x))
Eq(F_66(x), 2*F_18(x) + F_58(x) + F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_64(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 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