0132_0213_1032_1230_1320
Counting sequence:
1, 1, 2, 6, 19, 55, 150, 395, 1021, 2616, 6683, 17079, 43738, 112327, 289341, 747428, 1935655, 5023743, 13062014, 34011987, 88668157, 231372544, 604196067, 1578686951, 4126771954, 10791385935, 28226807165, 73847728140, 193233519631, 505686722671, 1323493796598, 3464127935099, 9067554880573, 23735863757832, 62134686138731, 162657487100823, 425816348642890, 1114748687331223, 2918343940496829, 7640111540135732, 20001647413686967, 52364144041986399, 137089411189726382, 358901342150940771, 939609119974763389, 2459915026329695440, 6440113974724197555, 16860382926992840135, 44140946860881584866, 115562281758964008351, 302545546613019574013, 792073654458555497244, 2073674009494395946783, 5428945559447400379087, 14213157039627438702246, 37210514300887550168843, 97418363345767945544125, 255044530701602586218328, 667715138688960709753019, 1748100705224390465732727, 4576586616701249959369402, 11981658424311524512171879, 31368387215094557759227197, 82123500338689607668107716, 215002108036401077571216199, 562882812241354134757141695, 1473646305629321125703695070, 3858056058529894505000560755, 10100521777726877358811151101, 26443509090183677619318794656, 69230005123889890147996540803, 181246505543617226784312521639, 474509510031223877339242993618, 1242282021598577963378978057967, 3252336548861556132180790619261, 8514727613180181058898526578988, 22291846267067165686045964818735, 58360811140797669059331563323183, 152790587060878540779223870229334, 400010949852943340797010935752347, 1047242262120162238760362066219837, 2741715835751964860836513471938984, 7177905243624575267620598070725387, 18791999892099446713989995436205911, 49198094426629136295667236186606634, 128802283375698704817257702831531575, 337208755676288463123596564091089085, 882823983604809653969122379185692500, 2311263195041426436774559071498922711, 6050965601326041530976348556099719711, 15841633608549841903197898481645320718, 41473935223549771669144211826516982659, 108580172060552048079528332484525475453, 284266580955011522510707883379548079472, 744219570798292819320049285549603341651, 1948392131427487535159947588229987349383, 5100956823459410985541328160622073067970, 13354478338901227820163225387329680377663, 34962478193145237272243363571044148465149, 91532956240336413590989654178177428972092, 239636390527467862689300572963548052867647
Generating function in Maple syntax:
-(x^7+x^6-5*x^4-6*x^3+12*x^2-6*x+1)/(x-1)/(2*x-1)/(x^2-3*x+1)/(x^2+x-1)
Generating function in latex syntax:
-\frac{x^{7}+x^{6}-5 x^{4}-6 x^{3}+12 x^{2}-6 x +1}{\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right)}
Generating function in sympy syntax:
(-x**7 - x**6 + 5*x**4 + 6*x**3 - 12*x**2 + 6*x - 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)*(x^2-3*x+1)*(x^2+x-1)*F(x)+x^7+x^6-5*x^4-6*x^3+12*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+x^{7}+x^{6}-5 x^{4}-6 x^{3}+12 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,-208*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)+1034/5*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+208*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)-1034/5*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+5/4*2^n-415*(3/2-1/2*5^(1/2))^(1-n)-415*(1/2*5^(1/2)-1/2)^(1-n)-415*(3/2+1/2*5^(1/2))^(1-n)-415*(-1/2*5^(1/2)-1/2)^(1-n)+624*(3/2-1/2*5^(1/2))^(-n)-209*(1/2*5^(1/2)-1/2)^(-n)+624*(3/2+1/2*5^(1/2))^(-n)-209*(-1/2*5^(1/2)-1/2)^(-n)+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ -208 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}+\frac{1034 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+208 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}-\\\frac{1034 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+\frac{5 \,2^{n}}{4}-415 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}-415 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{1-n}-\\415 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n}-415 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{1-n}+624 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}-\\209 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+624 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}-209 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+2 & \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) = 55
a(6) = 150
a(7) = 395
a(n+5) = -2*a(n)+5*a(n+1)+4*a(n+2)-11*a(n+3)+6*a(n+4)-2, 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) = 55
a \! \left(6\right) = 150
a \! \left(7\right) = 395
a \! \left(n +5\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)+4 a \! \left(n +2\right)-11 a \! \left(n +3\right)+6 a \! \left(n +4\right)-2, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18337/
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[12,x]*F[4,x]
F[4,x] = F[23,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[21,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[14,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[13,x]
F[23,x] = F[24,x]+F[35,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[24,x]+F[28,x]
F[28,x] = F[29,x]+F[30,x]+F[34,x]
F[29,x] = 0
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[12,x]
F[33,x] = F[28,x]
F[34,x] = F[12,x]*F[24,x]
F[35,x] = F[29,x]+F[36,x]+F[58,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[44,x]
F[38,x] = F[10,x]+F[39,x]
F[39,x] = F[29,x]+F[40,x]+F[41,x]
F[40,x] = F[10,x]*F[12,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[12,x]+F[43,x]
F[43,x] = F[41,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[29,x]+F[36,x]+F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[24,x]+F[45,x]
F[48,x] = F[29,x]+F[49,x]+F[54,x]+F[55,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[40,x]
F[52,x] = F[53,x]
F[53,x] = 2*F[29,x]+F[49,x]+F[54,x]
F[54,x] = F[12,x]*F[45,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[28,x]+F[57,x]
F[57,x] = F[55,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[47,x]+F[60,x]
F[60,x] = F[61,x]+F[67,x]
F[61,x] = F[29,x]+F[30,x]+F[62,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[27,x]+F[64,x]
F[64,x] = F[61,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[61,x]
F[67,x] = F[68,x]
F[68,x] = F[12,x]*F[60,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{23}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{21}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{35}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right)
F_{33}\! \left(x \right) = F_{28}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)+F_{58}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{44}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{29}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{24}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{29}\! \left(x \right)+F_{49}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \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_{40}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{28}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{47}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{67}\! \left(x \right)
F_{61}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{27}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{61}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_23(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_14(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_13(x))
Eq(F_23(x), F_24(x) + F_35(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_24(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_30(x) + F_34(x))
Eq(F_29(x), 0)
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_12(x))
Eq(F_33(x), F_28(x))
Eq(F_34(x), F_12(x)*F_24(x))
Eq(F_35(x), F_29(x) + F_36(x) + F_58(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_44(x))
Eq(F_38(x), F_10(x) + F_39(x))
Eq(F_39(x), F_29(x) + F_40(x) + F_41(x))
Eq(F_40(x), F_10(x)*F_12(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_12(x) + F_43(x))
Eq(F_43(x), F_41(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_29(x) + F_36(x) + F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_24(x) + F_45(x))
Eq(F_48(x), F_29(x) + F_49(x) + F_54(x) + F_55(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_40(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), 2*F_29(x) + F_49(x) + F_54(x))
Eq(F_54(x), F_12(x)*F_45(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_28(x) + F_57(x))
Eq(F_57(x), F_55(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_47(x) + F_60(x))
Eq(F_60(x), F_61(x) + F_67(x))
Eq(F_61(x), F_29(x) + F_30(x) + F_62(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_27(x) + F_64(x))
Eq(F_64(x), F_61(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_61(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x)*F_60(x))
Pack JSON:
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Specification JSON:
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