0132_0213_1023_1302_3120
Counting sequence:
1, 1, 2, 6, 19, 50, 114, 234, 442, 779, 1295, 2049, 3109, 4552, 6464, 8940, 12084, 16009, 20837, 26699, 33735, 42094, 51934, 63422, 76734, 92055, 109579, 129509, 152057, 177444, 205900, 237664, 272984, 312117, 355329, 402895, 455099, 512234, 574602, 642514, 716290, 796259, 882759, 976137, 1076749, 1184960, 1301144, 1425684, 1558972, 1701409, 1853405, 2015379, 2187759, 2370982, 2565494, 2771750, 2990214, 3221359, 3465667, 3723629, 3995745, 4282524, 4584484, 4902152, 5236064, 5586765, 5954809, 6340759, 6745187, 7168674, 7611810, 8075194, 8559434, 9065147, 9592959, 10143505, 10717429, 11315384, 11938032, 12586044, 13260100, 13960889, 14689109, 15445467, 16230679, 17045470, 17890574, 18766734, 19674702, 20615239, 21589115, 22597109, 23640009, 24718612, 25833724, 26986160, 28176744, 29406309, 30675697, 31985759, 33337355
Generating function in Maple syntax:
-(x^8+2*x^7+3*x^6-x^5+4*x^4-4*x^3+7*x^2-4*x+1)/(x-1)^5
Generating function in latex syntax:
-\frac{x^{8}+2 x^{7}+3 x^{6}-x^{5}+4 x^{4}-4 x^{3}+7 x^{2}-4 x +1}{\left(x -1\right)^{5}}
Generating function in sympy syntax:
(-x**8 - 2*x**7 - 3*x**6 + x**5 - 4*x**4 + 4*x**3 - 7*x**2 + 4*x - 1)/(x - 1)**5
Implicit equation for the generating function in Maple syntax:
(x-1)^5*F(x)+x^8+2*x^7+3*x^6-x^5+4*x^4-4*x^3+7*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{5} F \! \left(x \right)+x^{8}+2 x^{7}+3 x^{6}-x^{5}+4 x^{4}-4 x^{3}+7 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,80+209/8*n^2-877/12*n-53/12*n^3+3/8*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 80+\frac{209}{8} n^{2}-\frac{877}{12} n -\frac{53}{12} n^{3}+\frac{3}{8} n^{4} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 50
a(6) = 114
a(7) = 234
a(8) = 442
a(n) = 80+209/8*n^2-877/12*n-53/12*n^3+3/8*n^4, 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) = 50
a \! \left(6\right) = 114
a \! \left(7\right) = 234
a \! \left(8\right) = 442
a \! \left(n \right) = 80+\frac{209}{8} n^{2}-\frac{877}{12} n -\frac{53}{12} n^{3}+\frac{3}{8} n^{4}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18310/
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[20,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[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[13,x]*F[4,x]
F[20,x] = F[21,x]+F[40,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[1,x]+F[14,x]
F[25,x] = F[21,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[39,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[1,x]+F[30,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]+F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[36,x]*F[4,x]
F[39,x] = F[27,x]
F[40,x] = F[41,x]+F[42,x]+F[66,x]
F[41,x] = 0
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[53,x]
F[44,x] = F[30,x]+F[45,x]
F[45,x] = F[41,x]+F[46,x]+F[50,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[36,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[14,x]*F[4,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[14,x]+F[52,x]
F[52,x] = F[50,x]
F[53,x] = F[54,x]+F[58,x]
F[54,x] = F[41,x]+F[42,x]+F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[21,x]+F[57,x]
F[57,x] = F[55,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[65,x]
F[61,x] = F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[30,x]+F[62,x]
F[65,x] = F[59,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[84,x]
F[68,x] = F[21,x]+F[69,x]
F[69,x] = F[41,x]+F[70,x]+F[83,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[74,x]
F[72,x] = F[4,x]+F[73,x]
F[73,x] = F[41,x]+F[46,x]+F[49,x]
F[74,x] = F[75,x]+F[77,x]
F[75,x] = F[41,x]+F[70,x]+F[76,x]
F[76,x] = F[21,x]*F[4,x]
F[77,x] = F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]+F[82,x]
F[80,x] = F[81,x]
F[81,x] = F[30,x]*F[4,x]
F[82,x] = F[78,x]
F[83,x] = F[4,x]*F[68,x]
F[84,x] = F[85,x]+F[90,x]
F[85,x] = F[86,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[88,x]+F[89,x]
F[88,x] = F[21,x]
F[89,x] = F[85,x]
F[90,x] = F[91,x]
F[91,x] = F[4,x]*F[84,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_{20}\! \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_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{40}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{39}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \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_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{27}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{66}\! \left(x \right)
F_{41}\! \left(x \right) = 0
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{53}\! \left(x \right)
F_{44}\! \left(x \right) = F_{30}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{36}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{14}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{54}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \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_{21}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{55}\! \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_{65}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \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_{30}\! \left(x \right)+F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{59}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{84}\! \left(x \right)
F_{68}\! \left(x \right) = F_{21}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{41}\! \left(x \right)+F_{70}\! \left(x \right)+F_{83}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{74}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{77}\! \left(x \right)
F_{75}\! \left(x \right) = F_{41}\! \left(x \right)+F_{70}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{82}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{82}\! \left(x \right) = F_{78}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)+F_{90}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)+F_{89}\! \left(x \right)
F_{88}\! \left(x \right) = F_{21}\! \left(x \right)
F_{89}\! \left(x \right) = F_{85}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \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_20(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_14(x) + F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_13(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_40(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_1(x) + F_14(x))
Eq(F_25(x), F_21(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_39(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_1(x) + F_30(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x) + F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_36(x)*F_4(x))
Eq(F_39(x), F_27(x))
Eq(F_40(x), F_41(x) + F_42(x) + F_66(x))
Eq(F_41(x), 0)
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_53(x))
Eq(F_44(x), F_30(x) + F_45(x))
Eq(F_45(x), F_41(x) + F_46(x) + F_50(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_36(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_14(x)*F_4(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_14(x) + F_52(x))
Eq(F_52(x), F_50(x))
Eq(F_53(x), F_54(x) + F_58(x))
Eq(F_54(x), F_41(x) + F_42(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_21(x) + F_57(x))
Eq(F_57(x), F_55(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) + F_65(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_30(x) + F_62(x))
Eq(F_65(x), F_59(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_84(x))
Eq(F_68(x), F_21(x) + F_69(x))
Eq(F_69(x), F_41(x) + F_70(x) + F_83(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_74(x))
Eq(F_72(x), F_4(x) + F_73(x))
Eq(F_73(x), F_41(x) + F_46(x) + F_49(x))
Eq(F_74(x), F_75(x) + F_77(x))
Eq(F_75(x), F_41(x) + F_70(x) + F_76(x))
Eq(F_76(x), F_21(x)*F_4(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_82(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_30(x)*F_4(x))
Eq(F_82(x), F_78(x))
Eq(F_83(x), F_4(x)*F_68(x))
Eq(F_84(x), F_85(x) + F_90(x))
Eq(F_85(x), F_86(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_88(x) + F_89(x))
Eq(F_88(x), F_21(x))
Eq(F_89(x), F_85(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_4(x)*F_84(x))
Pack JSON:
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Specification JSON:
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