0132_0321_1023_1230_2103
Counting sequence:
1, 1, 2, 6, 19, 47, 104, 235, 534, 1204, 2704, 6077, 13661, 30700, 68981, 155000, 348288, 782600, 1758485, 3951281, 8878454, 19949709, 44826588, 100724430, 226325746, 508549339, 1142699991, 2567623574, 5769397807, 12963719202, 29129212634, 65452746662, 147070986763, 330465507559, 742549255216, 1668493031227, 3749069810118, 8424083396252, 18928743571392, 42532500728917, 95569661632981, 214743080423492, 482523321751045, 1084220062292600, 2436220365912760, 5474137472367080, 12300275248354317, 27638467603162953, 62103072982313150, 139544338319434941, 313553282018020076, 704547829373161942, 1583104602444909026, 3557203752244959923, 7992964277561666927, 17960027704923384750, 40355815935163476511, 90678695297688670850, 203753178825617433458, 457829237013760061254, 1028732957555448885123, 2311541973299040398047, 5193987667139769619960, 11670784350023130752843, 26224014393886990727606, 58924825470657342588100, 132402880985178545150960, 297506573047127442162413, 668491201608776086887693, 1502086095279501070786844, 3375156819120650786298965, 7583908531912026556497080, 17040887787665202828506288, 38290527288121781427210696, 86038033831996739126430597, 193325707164450056851565601, 434398920872775071402351110, 976085515078003460529837229, 2193244243864331935610459964, 4928175081933892260348406046, 11073508892654112995777434834, 24881948623377786316292815755, 55909231057475793368014660295, 125626901845675260056544701510, 282281086125448527164811247567, 634279843039096521018152536354, 1425213870357966309144571618762, 3202426497629580612142484526310, 7195787022578031012411388135035, 16168786672427676327820689177623, 36330933869803803055910281963968, 81634867389457251427229864970523, 183431881976290629582549322727398, 412167697472234707536418228461356, 926132409531302793228155914679584, 2081000634558549664410180735093125, 4675965981176167414512099156404485, 10506800187379581700206223133222516, 23608565721376781150514364687756389, 53047965648956976586722853352330808, 119197696831911152623753848259195496
Generating function in Maple syntax:
-(x+1)*(x^6-5*x^5+7*x^4-3*x^3+4*x^2-3*x+1)/(x-1)/(x^2+1)/(x^3-x^2-2*x+1)
Generating function in latex syntax:
-\frac{\left(x +1\right) \left(x^{6}-5 x^{5}+7 x^{4}-3 x^{3}+4 x^{2}-3 x +1\right)}{\left(x -1\right) \left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right)}
Generating function in sympy syntax:
(-x - 1)*(x**6 - 5*x**5 + 7*x**4 - 3*x**3 + 4*x**2 - 3*x + 1)/((x - 1)*(x**2 + 1)*(x**3 - x**2 - 2*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2+1)*(x^3-x^2-2*x+1)*F(x)+(x+1)*(x^6-5*x^5+7*x^4-3*x^3+4*x^2-3*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right) F \! \left(x \right)+\left(x +1\right) \left(x^{6}-5 x^{5}+7 x^{4}-3 x^{3}+4 x^{2}-3 x +1\right) = 0
Explicit closed form in Maple syntax:
3407/2366*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^6-2*_Z^5+_Z^3-2*_Z^2+3*_Z-1))-283/169*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^6-2*_Z^5+_Z^3-2*_Z^2+3*_Z-1))-3209/2366*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^6-2*_Z^5+_Z^3-2*_Z^2+3*_Z-1))+1753/2366*Sum(_alpha^(1-n),_alpha = RootOf(_Z^6-2*_Z^5+_Z^3-2*_Z^2+3*_Z-1))-6525/2366*Sum(_alpha^(-n),_alpha = RootOf(_Z^6-2*_Z^5+_Z^3-2*_Z^2+3*_Z-1))+1902/1183*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^6-2*_Z^5+_Z^3-2*_Z^2+3*_Z-1))-piecewise(n = 0,-2,n = 1,1,0)
Explicit closed form in latex syntax:
\frac{3407 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{3}-2 \textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{2366}-\frac{283 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{3}-2 \textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{169}-\frac{3209 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{3}-2 \textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{2366}+\frac{1753 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{3}-2 \textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{2366}-\frac{6525 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{3}-2 \textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{2366}+\frac{1902 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{3}-2 \textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{1183}-\left(\left\{\begin{array}{cc}-2 & n =0 \\ 1 & n =1 \\ 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) = 47
a(6) = 104
a(7) = 235
a(n+2) = a(n)-a(n+1)-2*a(n+4)+a(n+5)-4, 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) = 47
a \! \left(6\right) = 104
a \! \left(7\right) = 235
a \! \left(n +2\right) = a \! \left(n \right)-a \! \left(n +1\right)-2 a \! \left(n +4\right)+a \! \left(n +5\right)-4, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19310/
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[63,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[37,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[1,x]+F[4,x]
F[30,x] = F[15,x]+F[31,x]+F[35,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[17,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[11,x]*F[4,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[11,x]+F[18,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[2,x]*F[4,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[15,x]+F[47,x]+F[54,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[51,x]
F[49,x] = F[4,x]+F[50,x]
F[50,x] = F[15,x]+F[31,x]+F[34,x]
F[51,x] = F[41,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[46,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[2,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[17,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[55,x]+F[65,x]
F[65,x] = F[46,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]
F[69,x] = x^2
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_{63}\! \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_{37}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{17}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \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_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{54}\! \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_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{51}\! \left(x \right) = F_{41}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{2}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{17}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{55}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{46}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \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_{69}\! \left(x \right)
F_{69}\! \left(x \right) = x^{2}
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_63(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_37(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_1(x) + F_4(x))
Eq(F_30(x), F_15(x) + F_31(x) + F_35(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_17(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_11(x)*F_4(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_11(x) + F_18(x))
Eq(F_37(x), F_38(x) + F_43(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_2(x)*F_4(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_15(x) + F_47(x) + F_54(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_51(x))
Eq(F_49(x), F_4(x) + F_50(x))
Eq(F_50(x), F_15(x) + F_31(x) + F_34(x))
Eq(F_51(x), F_41(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_46(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_2(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_17(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_55(x) + F_65(x))
Eq(F_65(x), F_46(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), x**2)
Pack JSON:
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Specification JSON:
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