0132_0321_1023_1032_1230
Counting sequence:
1, 1, 2, 6, 19, 53, 139, 357, 901, 2246, 5553, 13646, 33385, 81409, 198029, 480825, 1165855, 2823894, 6834535, 16531390, 39967991, 96597365, 233401815, 563840757, 1361892177, 3289112814, 7942854117, 19179853910, 46311818813, 111820517585, 269984169769, 651846455833, 1573783021979, 3799607354830, 9173356125931, 22146978796566, 53468526158267, 129086261136469, 311645150083651, 752384105418789, 1816427236712061, 4385264100401206, 10587002378978617, 25559355197171710, 61705871575157745, 148971390429600321, 359649189657120485, 868270757850552217, 2096192522769811063, 5060659146131233510, 12217516963291634319, 29495704381126503694, 71208946524957058655, 171913635687124395733, 415036288263114043343, 1001986341632756867317, 2419009209568024130825, 5840005198591514059934, 14099020412032561919405, 34038047503800252851270, 82175118143880901174149, 198388288802235013373009, 478951704964415334598337, 1156291715682050880973497, 2791535167506239659800755, 6739362108039302368912414, 16270259489058324327622227, 39279881280151925685746342, 94830022406176402459038435, 228939926748788411955332853, 552709877110847109142726779, 1334359683190674421125242213, 3221429247575765306402199285, 7777218185853060062596107302, 18775865633096501606154324289, 45328949477455103833140120878, 109433764634741221033896307961, 264196478832895714395189754113, 637826722458634370638229940413, 1539849924040958856741322518393, 3717526571075406374498759002191, 8974903067175522018000351555446, 21667332707235849514208530054199, 52309568484975224852765874662622, 126286469683307452542059321352007, 304882507862848686169261090279093, 736051485429712538241625579793991, 1776985478760361185568251942635893, 4290022443020488601887289808630849, 10357030364930187218128775674115278, 25004083173117852982355685307412981, 60365196711601785648346094897275574, 145734476597123155517550611975067725, 351834149907322710331666848479400529, 849402776414480813533112524047296217, 2050639702741272919636842378192513497, 4950682181906202086046195506796260683, 11952004066570553344559810314880920302, 28854690315078349043474741767636444731, 69661384696784343385888194630691976118, 168177459708752044290769534363732792203
Generating function in Maple syntax:
-(x^8-x^7-5*x^6-2*x^5+4*x^4+3*x^3+x^2-3*x+1)/(x-1)/(x^2+2*x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{x^{8}-x^{7}-5 x^{6}-2 x^{5}+4 x^{4}+3 x^{3}+x^{2}-3 x +1}{\left(x -1\right) \left(x^{2}+2 x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(-x**8 + x**7 + 5*x**6 + 2*x**5 - 4*x**4 - 3*x**3 - x**2 + 3*x - 1)/((x - 1)*(x**2 + 2*x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2+2*x-1)*(x^3+x^2+x-1)*F(x)+x^8-x^7-5*x^6-2*x^5+4*x^4+3*x^3+x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}+2 x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{8}-x^{7}-5 x^{6}-2 x^{5}+4 x^{4}+3 x^{3}+x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
555/352*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))+1337/352*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))+41/88*Sum(_alpha^(2-n),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))-581/176*Sum(_alpha^(1-n),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))-2139/352*Sum(_alpha^(-n),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))+1157/352*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))-piecewise(n = 0,2,n = 1,-3,n = 2,1,0)
Explicit closed form in latex syntax:
\frac{555 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{352}+\frac{1337 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{352}+\frac{41 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{88}-\frac{581 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{176}-\frac{2139 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{352}+\frac{1157 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{352}-\left(\left\{\begin{array}{cc}2 & n =0 \\ -3 & n =1 \\ 1 & n =2 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 139
a(7) = 357
a(8) = 901
a(n+2) = -1/2*a(n)-3/2*a(n+1)+3/2*a(n+4)-1/2*a(n+5)-1/2, n >= 9
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) = 53
a \! \left(6\right) = 139
a \! \left(7\right) = 357
a \! \left(8\right) = 901
a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{2}-\frac{3 a \! \left(n +1\right)}{2}+\frac{3 a \! \left(n +4\right)}{2}-\frac{a \! \left(n +5\right)}{2}-\frac{1}{2}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19299/
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[22,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[16,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]+F[21,x]
F[21,x] = F[16,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[15,x]+F[24,x]+F[65,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[36,x]
F[26,x] = F[27,x]+F[4,x]
F[27,x] = F[15,x]+F[28,x]+F[32,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[17,x]+F[30,x]
F[30,x] = F[31,x]+F[33,x]
F[31,x] = F[15,x]+F[16,x]+F[32,x]
F[32,x] = F[11,x]*F[4,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[18,x]
F[36,x] = F[37,x]+F[39,x]
F[37,x] = F[15,x]+F[24,x]+F[38,x]
F[38,x] = F[2,x]*F[4,x]
F[39,x] = F[15,x]+F[40,x]+F[41,x]+F[60,x]
F[40,x] = 0
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[59,x]
F[43,x] = F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[53,x]
F[47,x] = F[4,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[17,x]+F[51,x]
F[51,x] = F[18,x]+F[52,x]
F[52,x] = F[34,x]
F[53,x] = F[44,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[43,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[43,x]
F[59,x] = F[58,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[15,x]+F[24,x]+F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[2,x]+F[64,x]
F[64,x] = F[45,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[63,x]+F[67,x]
F[67,x] = F[61,x]+F[68,x]
F[68,x] = F[58,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_{22}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \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_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
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_{21}\! \left(x \right) = F_{16}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{65}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{17}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{33}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \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) = F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \left(x \right)+F_{60}\! \left(x \right)
F_{40}\! \left(x \right) = 0
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{59}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{53}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \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_{17}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{18}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{34}\! \left(x \right)
F_{53}\! \left(x \right) = F_{44}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{43}\! \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_{43}\! \left(x \right)
F_{59}\! \left(x \right) = F_{58}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{2}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{45}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{63}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{61}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{58}\! \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_22(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x) + F_21(x))
Eq(F_21(x), F_16(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_15(x) + F_24(x) + F_65(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_36(x))
Eq(F_26(x), F_27(x) + F_4(x))
Eq(F_27(x), F_15(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_17(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_33(x))
Eq(F_31(x), F_15(x) + F_16(x) + F_32(x))
Eq(F_32(x), F_11(x)*F_4(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_18(x))
Eq(F_36(x), F_37(x) + F_39(x))
Eq(F_37(x), F_15(x) + F_24(x) + F_38(x))
Eq(F_38(x), F_2(x)*F_4(x))
Eq(F_39(x), F_15(x) + F_40(x) + F_41(x) + F_60(x))
Eq(F_40(x), 0)
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_59(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_53(x))
Eq(F_47(x), F_4(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_17(x) + F_51(x))
Eq(F_51(x), F_18(x) + F_52(x))
Eq(F_52(x), F_34(x))
Eq(F_53(x), F_44(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_43(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_43(x))
Eq(F_59(x), F_58(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_15(x) + F_24(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_2(x) + F_64(x))
Eq(F_64(x), F_45(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_63(x) + F_67(x))
Eq(F_67(x), F_61(x) + F_68(x))
Eq(F_68(x), F_58(x))
Pack JSON:
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Specification JSON:
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