0123_0132_0321_1023_2103
Counting sequence:
1, 1, 2, 6, 19, 51, 133, 356, 968, 2623, 7076, 19084, 51524, 139150, 375730, 1014417, 2738829, 7394826, 19966068, 53908083, 145550629, 392983951, 1061050244, 2864818086, 7734959455, 20884256394, 56387133052, 152244292718, 411056977574, 1109846777008, 2996567230043, 8090680052959, 21844697188190, 58980307211284, 159245816416969, 429960969118827, 1160887231672899, 3134375586288906, 8462760247295208, 22849307312193117, 61692737285707822, 166569331043913820, 449734332842406170, 1214274974087497416, 3278521573783913986, 8851951937694340779, 23900118191632633435, 64529908611634859426, 174229644892875178634, 470417048661745502653, 1270117951555769277509, 3429296653795809045733, 9259042064030503608962, 24999254540604945911050, 67497557875215864341869, 182242246932530686760904, 492050936545254587737188, 1328529077259989533357932, 3587005689934198037250888, 9684853752811276649376612, 26148938786618796972703981, 70601685592604436753985009, 190623338453327157124575950, 514679739698708589560551962, 1389626457104507420240293771, 3751967566112590717561119039, 10130247985126086091271879953, 27351495563826950857882512932, 73848568236084903216511365044, 199389865822629001995852524155, 538349213022923817962220451988, 1453533628535645427656641934464, 3924515831314189701148545527432, 10596125337500577857550850605286, 28609356413380453469437169490694, 77244770925001211359667549123421, 208559554749903114915483212093541, 563107215629006059052570651066262, 1520379810331366406911213809079872, 4104999374020047338594834016971307, 11083427802841123530492940285770961, 29925064699946942585272829207504059, 80797160700271434294035969151994556, 218150946127680761731864428535626402, 589003807608333421290797413998274627, 1590300163878564685183508768670737610, 4293783127653261354136921548973551856, 11593140695122031050283750821413819746, 31301280754333406332614566709636387062, 84512920409380893701011664256071003860, 228183433520800520903290819511003461919, 616091351252868352314325011845800482487, 1663436066466169964074442436751703655722, 4491248808465661340604716402714298805936, 12126294641666924401409543322184062429337, 32740787252611689106507673907009796719687, 88399563229928634513496180279759709995623, 238677302379734310348049026677995695182658, 644424616930464387792275204068907929727364, 1739935397146657039007184052515550706073633, 4697795687358975076125067509192606886397926
Generating function in Maple syntax:
-(x^2+x-1)/(x^8+3*x^7+3*x^6-x^5-4*x^4-x^3-x^2-2*x+1)
Generating function in latex syntax:
-\frac{x^{2}+x -1}{x^{8}+3 x^{7}+3 x^{6}-x^{5}-4 x^{4}-x^{3}-x^{2}-2 x +1}
Generating function in sympy syntax:
(-x**2 - x + 1)/(x**8 + 3*x**7 + 3*x**6 - x**5 - 4*x**4 - x**3 - x**2 - 2*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^8+3*x^7+3*x^6-x^5-4*x^4-x^3-x^2-2*x+1)*F(x)+x^2+x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{8}+3 x^{7}+3 x^{6}-x^{5}-4 x^{4}-x^{3}-x^{2}-2 x +1\right) F \! \left(x \right)+x^{2}+x -1 = 0
Explicit closed form in Maple syntax:
3217899/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+6)+3217899/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+6)+3217899/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+6)+3217899/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+6)+3217899/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+6)+3217899/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+6)+3217899/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+6)+3217899/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+6)+3843659/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+5)+3843659/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+5)+3843659/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+5)+3843659/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+5)+3843659/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+5)+3843659/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+5)+3843659/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+5)+3843659/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+5)-970296/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+4)-970296/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+4)-970296/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+4)-970296/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+4)-970296/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+4)-970296/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+4)-970296/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+4)-970296/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+4)-12994139/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+3)-12994139/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+3)-12994139/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+3)-12994139/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+3)-12994139/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+3)-12994139/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+3)-12994139/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+3)-12994139/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+3)-2762665/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+2)-2762665/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+2)-2762665/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+2)-2762665/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+2)-2762665/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+2)-2762665/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+2)-2762665/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+2)-2762665/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+2)+3691396/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+1)+3691396/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+1)+3691396/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+1)+3691396/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+1)+3691396/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+1)+3691396/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+1)+3691396/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+1)+3691396/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+1)+6280513/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 1)^(-n-1)+6280513/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 2)^(-n-1)+6280513/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 3)^(-n-1)+6280513/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 4)^(-n-1)+6280513/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 5)^(-n-1)+6280513/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 6)^(-n-1)+6280513/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 7)^(-n-1)+6280513/61270354*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 8)^(-n-1)+1383910/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 1)^(-n)+1383910/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 2)^(-n)+1383910/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 3)^(-n)+1383910/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 4)^(-n)+1383910/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 5)^(-n)+1383910/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 6)^(-n)+1383910/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 7)^(-n)+1383910/30635177*RootOf(_Z^8+3*_Z^7+3*_Z^6-_Z^5-4*_Z^4-_Z^3-_Z^2-2*_Z+1,index = 8)^(-n)
Explicit closed form in latex syntax:
\frac{3217899 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{61270354}+\frac{3217899 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{61270354}+\frac{3217899 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{61270354}+\frac{3217899 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{61270354}+\frac{3217899 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{61270354}+\frac{3217899 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{61270354}+\frac{3217899 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{61270354}+\frac{3217899 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{61270354}+\frac{3843659 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{61270354}+\frac{3843659 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{61270354}+\frac{3843659 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{61270354}+\frac{3843659 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{61270354}+\frac{3843659 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{61270354}+\frac{3843659 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{61270354}+\frac{3843659 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{61270354}+\frac{3843659 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{61270354}-\frac{970296 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{30635177}-\frac{970296 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{30635177}-\frac{970296 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{30635177}-\frac{970296 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{30635177}-\frac{970296 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{30635177}-\frac{970296 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{30635177}-\frac{970296 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{30635177}-\frac{970296 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{30635177}-\frac{12994139 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{61270354}-\frac{12994139 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{61270354}-\frac{12994139 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{61270354}-\frac{12994139 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{61270354}-\frac{12994139 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{61270354}-\frac{12994139 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{61270354}-\frac{12994139 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{61270354}-\frac{12994139 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{61270354}-\frac{2762665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{30635177}-\frac{2762665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{30635177}-\frac{2762665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{30635177}-\frac{2762665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{30635177}-\frac{2762665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{30635177}-\frac{2762665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{30635177}-\frac{2762665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{30635177}-\frac{2762665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{30635177}+\frac{3691396 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{30635177}+\frac{3691396 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{30635177}+\frac{3691396 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{30635177}+\frac{3691396 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{30635177}+\frac{3691396 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{30635177}+\frac{3691396 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{30635177}+\frac{3691396 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{30635177}+\frac{3691396 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{30635177}+\frac{6280513 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{61270354}+\frac{6280513 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{61270354}+\frac{6280513 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{61270354}+\frac{6280513 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{61270354}+\frac{6280513 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{61270354}+\frac{6280513 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{61270354}+\frac{6280513 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{61270354}+\frac{6280513 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{61270354}+\frac{1383910 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{30635177}+\frac{1383910 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{30635177}+\frac{1383910 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{30635177}+\frac{1383910 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{30635177}+\frac{1383910 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{30635177}+\frac{1383910 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{30635177}+\frac{1383910 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{30635177}+\frac{1383910 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+3 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}-\textit{\_Z}^{5}-4 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{30635177}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 133
a(7) = 356
a(n+8) = -a(n)-3*a(n+1)-3*a(n+2)+a(n+3)+4*a(n+4)+a(n+5)+a(n+6)+2*a(n+7), 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) = 19
a \! \left(5\right) = 51
a \! \left(6\right) = 133
a \! \left(7\right) = 356
a \! \left(n +8\right) = -a \! \left(n \right)-3 a \! \left(n +1\right)-3 a \! \left(n +2\right)+a \! \left(n +3\right)+4 a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16215/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[59,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[33,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[4,x]
F[26,x] = F[15,x]+F[27,x]+F[30,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[17,x]+F[29,x]
F[29,x] = F[18,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[12,x]+F[32,x]
F[32,x] = F[16,x]
F[33,x] = F[34,x]+F[39,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[2,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[2,x]*F[4,x]
F[39,x] = 2*F[15,x]+F[40,x]+F[55,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[37,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[15,x]+F[46,x]+F[52,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[50,x]
F[48,x] = F[4,x]+F[49,x]
F[49,x] = F[15,x]+F[18,x]+F[27,x]
F[50,x] = F[37,x]+F[51,x]
F[51,x] = 2*F[15,x]+F[40,x]+F[44,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[2,x]+F[54,x]
F[54,x] = F[15,x]+F[38,x]+F[46,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[45,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[42,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[53,x]+F[61,x]
F[61,x] = F[45,x]+F[62,x]
F[62,x] = F[15,x]+F[44,x]+F[58,x]+F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[70,x]
F[65,x] = F[66,x]+F[68,x]
F[66,x] = F[67,x]
F[67,x] = x^2
F[68,x] = F[69,x]
F[69,x] = F[32,x]*F[4,x]
F[70,x] = F[71,x]+F[73,x]
F[71,x] = F[72,x]
F[72,x] = F[37,x]*F[4,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[57,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{59}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right)+F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{39}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{2}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{55}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{37}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{52}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{50}\! \left(x \right) = F_{37}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{2}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{46}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{45}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{53}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{45}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{58}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{70}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = x^{2}
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_59(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_33(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_4(x))
Eq(F_26(x), F_15(x) + F_27(x) + F_30(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_17(x) + F_29(x))
Eq(F_29(x), F_18(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_12(x) + F_32(x))
Eq(F_32(x), F_16(x))
Eq(F_33(x), F_34(x) + F_39(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_2(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_2(x)*F_4(x))
Eq(F_39(x), 2*F_15(x) + F_40(x) + F_55(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_37(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_15(x) + F_46(x) + F_52(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_50(x))
Eq(F_48(x), F_4(x) + F_49(x))
Eq(F_49(x), F_15(x) + F_18(x) + F_27(x))
Eq(F_50(x), F_37(x) + F_51(x))
Eq(F_51(x), 2*F_15(x) + F_40(x) + F_44(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_2(x) + F_54(x))
Eq(F_54(x), F_15(x) + F_38(x) + F_46(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_45(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_42(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_53(x) + F_61(x))
Eq(F_61(x), F_45(x) + F_62(x))
Eq(F_62(x), F_15(x) + F_44(x) + F_58(x) + F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_70(x))
Eq(F_65(x), F_66(x) + F_68(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), x**2)
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_32(x)*F_4(x))
Eq(F_70(x), F_71(x) + F_73(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_37(x)*F_4(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_4(x)*F_57(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, 1, 3, 2], "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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 1, 3, 2], "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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 1, 3, 2], "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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [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, 0], [1, 2], [1, 2]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [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, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 0], [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": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [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, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 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