0213_1230_1320_3021
Counting sequence:
1, 1, 2, 6, 20, 62, 180, 503, 1375, 3708, 9912, 26340, 69710, 183959, 484449, 1273862, 3345940, 8781330, 23032416, 60384039, 158254955, 414649736, 1086229488, 2845105016, 7451211370, 19512769767, 51095561565, 133790813538, 350310629900, 917208502838, 2401449610572, 6287409596375, 16461317395735, 43097618510964, 112833689145384, 295407749595372, 773398158802790, 2024803921612535, 5301047989931001, 13878408806745374, 36334315932504580, 95124813971009466, 249040675901917848, 651998313514286151, 1706956464097690595, 4468875477526705472, 11699678765710351200, 30630178413626704448, 80190891662513066050, 209942566947464198343, 549636949925146626357, 1438968564315538660410, 3767269305991787787260, 9862840479592682824094, 25821254384639390661444, 67600927178011384102007, 176981536156733600248015, 463343699306813777557740, 1213049597792870186685336, 3175805166129982107156084, 8314366044713220932664830, 21767293256241604851306071, 56987514300474850655458833, 149195250798108504461455286, 390598240399700229412920820, 1022599475012688812414552802, 2677200193861755412365690960, 7009001125019349276282106983, 18349803218089825509469946075, 48040408603037176270426185848, 125771422738595773560515485392, 329273859907898239982963805320, 862050157575395064808602752410, 2256876613998879073614267634599, 5908579686782426203984555807629, 15468862451070767326178589930642, 40498007675874610851775582389740, 106025160595442534577081361000070, 277577474148231930270365363204796, 726707261924811128904323375824919, 1902544311777317198366767435784455, 4980925673709371944516364573672676, 13040232709955261582878703245730568, 34139772457365338685040165059836796, 89379084664558606210665903377947670, 233997481541146183385916705589198007, 612613359968551350763696743698418345, 1603842598383850682439203733028845518, 4198914435221686323461474226617328100, 10992900707358579541500642992370435978, 28779787687008794807731106700861084552, 75346462353977289894394089373308414855, 197259599375542044899753898166889252435, 516432335773886784851693099207004813104, 1352037407948594189746096320419045312064, 3539679888076847544563477657369595843152, 9267002256291851964290560128523925311090, 24261326880818515388988449521455424400775, 63516978386203308284015541748128968289381, 166289608277870637625707741912869731163754, 435351846447567060918355135661501868271580
Generating function in Maple syntax:
-(2*x^8+3*x^7-x^6-4*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{2 x^{8}+3 x^{7}-x^{6}-4 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:
(-2*x**8 - 3*x**7 + x**6 + 4*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)+2*x^8+3*x^7-x^6-4*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)+2 x^{8}+3 x^{7}-x^{6}-4 x^{4}-6 x^{3}+12 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/20*(-81*5^(1/2)+195)*(3/2-1/2*5^(1/2))^(-n)+1/20*(81*5^(1/2)+195)*(3/2+1/2*5^(1/2))^(-n)+1/20*(3*5^(1/2)-5)*(-1/2*5^(1/2)-1/2)^(-n)+1/20*(-3*5^(1/2)-5)*(1/2*5^(1/2)-1/2)^(-n)-2^(-1+n)-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-81 \sqrt{5}+195\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(81 \sqrt{5}+195\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\\\frac{\left(3 \sqrt{5}-5\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+\frac{\left(-3 \sqrt{5}-5\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}-2^{-1+n}-1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(6) = 180
a(7) = 503
a(8) = 1375
a(n+5) = -2*a(n)+5*a(n+1)+4*a(n+2)-11*a(n+3)+6*a(n+4)+1, 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) = 20
a \! \left(5\right) = 62
a \! \left(6\right) = 180
a \! \left(7\right) = 503
a \! \left(8\right) = 1375
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)+1, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15761/
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[14,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[18,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[14,x]
F[12,x] = F[13,x]+F[15,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = x
F[15,x] = F[10,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[10,x]*F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[14,x]*F[20,x]
F[20,x] = F[9,x]
F[21,x] = F[22,x]+F[33,x]
F[22,x] = F[23,x]
F[23,x] = F[14,x]*F[24,x]
F[24,x] = F[13,x]+F[25,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[32,x]
F[27,x] = 0
F[28,x] = F[14,x]*F[29,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[14,x]
F[31,x] = F[26,x]
F[32,x] = F[14,x]*F[22,x]
F[33,x] = F[27,x]+F[34,x]+F[72,x]
F[34,x] = F[14,x]*F[35,x]
F[35,x] = F[36,x]+F[51,x]
F[36,x] = F[37,x]+F[6,x]
F[37,x] = F[27,x]+F[38,x]+F[39,x]
F[38,x] = F[14,x]*F[6,x]
F[39,x] = F[14,x]*F[40,x]
F[40,x] = F[41,x]+F[45,x]
F[41,x] = F[14,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[14,x]*F[44,x]
F[44,x] = F[41,x]
F[45,x] = F[46,x]+F[48,x]
F[46,x] = F[47,x]
F[47,x] = x^2
F[48,x] = F[49,x]
F[49,x] = F[14,x]*F[50,x]
F[50,x] = F[45,x]
F[51,x] = F[33,x]+F[52,x]
F[52,x] = F[27,x]+F[53,x]+F[57,x]+F[58,x]
F[53,x] = F[14,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[37,x]
F[56,x] = F[52,x]
F[57,x] = F[14,x]*F[33,x]
F[58,x] = F[14,x]*F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[61,x]+F[65,x]
F[61,x] = F[62,x]
F[62,x] = F[14,x]*F[63,x]
F[63,x] = F[30,x]+F[64,x]
F[64,x] = F[61,x]
F[65,x] = 2*F[27,x]+F[66,x]+F[70,x]
F[66,x] = F[14,x]*F[67,x]
F[67,x] = F[68,x]+F[69,x]
F[68,x] = F[42,x]
F[69,x] = F[65,x]
F[70,x] = F[14,x]*F[71,x]
F[71,x] = F[60,x]
F[72,x] = F[14,x]*F[73,x]
F[73,x] = F[21,x]+F[74,x]
F[74,x] = F[75,x]+F[76,x]
F[75,x] = F[62,x]
F[76,x] = 2*F[27,x]+F[66,x]+F[77,x]
F[77,x] = F[14,x]*F[78,x]
F[78,x] = F[74,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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \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_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \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_{15}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{10}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{9}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{33}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{14}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)
F_{31}\! \left(x \right) = F_{26}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{22}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)+F_{72}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{51}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{6}\! \left(x \right)
F_{37}\! \left(x \right) = F_{27}\! \left(x \right)+F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right) F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{41}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = x^{2}
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{14}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{45}\! \left(x \right)
F_{51}\! \left(x \right) = F_{33}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{27}\! \left(x \right)+F_{53}\! \left(x \right)+F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{53}\! \left(x \right) = F_{14}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{37}\! \left(x \right)
F_{56}\! \left(x \right) = F_{52}\! \left(x \right)
F_{57}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{58}\! \left(x \right) = F_{14}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \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_{14}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{30}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{61}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{66}\! \left(x \right)+F_{70}\! \left(x \right)
F_{66}\! \left(x \right) = F_{14}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{69}\! \left(x \right)
F_{68}\! \left(x \right) = F_{42}\! \left(x \right)
F_{69}\! \left(x \right) = F_{65}\! \left(x \right)
F_{70}\! \left(x \right) = F_{14}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{60}\! \left(x \right)
F_{72}\! \left(x \right) = F_{14}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{21}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{62}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{66}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{14}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{74}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_21(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_14(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_18(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_14(x))
Eq(F_12(x), F_13(x) + F_15(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_10(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_10(x)*F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_14(x)*F_20(x))
Eq(F_20(x), F_9(x))
Eq(F_21(x), F_22(x) + F_33(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_14(x)*F_24(x))
Eq(F_24(x), F_13(x) + F_25(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_32(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_14(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_14(x))
Eq(F_31(x), F_26(x))
Eq(F_32(x), F_14(x)*F_22(x))
Eq(F_33(x), F_27(x) + F_34(x) + F_72(x))
Eq(F_34(x), F_14(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_51(x))
Eq(F_36(x), F_37(x) + F_6(x))
Eq(F_37(x), F_27(x) + F_38(x) + F_39(x))
Eq(F_38(x), F_14(x)*F_6(x))
Eq(F_39(x), F_14(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_45(x))
Eq(F_41(x), F_14(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_14(x)*F_44(x))
Eq(F_44(x), F_41(x))
Eq(F_45(x), F_46(x) + F_48(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), x**2)
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_14(x)*F_50(x))
Eq(F_50(x), F_45(x))
Eq(F_51(x), F_33(x) + F_52(x))
Eq(F_52(x), F_27(x) + F_53(x) + F_57(x) + F_58(x))
Eq(F_53(x), F_14(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_37(x))
Eq(F_56(x), F_52(x))
Eq(F_57(x), F_14(x)*F_33(x))
Eq(F_58(x), F_14(x)*F_59(x))
Eq(F_59(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_14(x)*F_63(x))
Eq(F_63(x), F_30(x) + F_64(x))
Eq(F_64(x), F_61(x))
Eq(F_65(x), 2*F_27(x) + F_66(x) + F_70(x))
Eq(F_66(x), F_14(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_69(x))
Eq(F_68(x), F_42(x))
Eq(F_69(x), F_65(x))
Eq(F_70(x), F_14(x)*F_71(x))
Eq(F_71(x), F_60(x))
Eq(F_72(x), F_14(x)*F_73(x))
Eq(F_73(x), F_21(x) + F_74(x))
Eq(F_74(x), F_75(x) + F_76(x))
Eq(F_75(x), F_62(x))
Eq(F_76(x), 2*F_27(x) + F_66(x) + F_77(x))
Eq(F_77(x), F_14(x)*F_78(x))
Eq(F_78(x), F_74(x))
Pack JSON:
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Specification JSON:
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