0132_0213_1302_3021_3102
Counting sequence:
1, 1, 2, 6, 19, 57, 162, 442, 1172, 3049, 7837, 20003, 50880, 129296, 328802, 837639, 2139043, 5477073, 14062782, 36203350, 93433136, 241671205, 626337577, 1626069311, 4227753084, 11005809692, 28680624062, 74805375747, 195248963887, 509918105709, 1332357869562, 3482659209058, 9106224516620, 23816418552481, 62302228951477, 163005442697819, 426538005482808, 1116243501530408, 2921436584864282, 7646502884966655, 20014842254629051, 52391358089677065, 137145488119054902, 359016793843072366, 939846619293890024, 2460403217271170077, 6441116741914998337, 16862441233125047927, 44145169018483524852, 115570937167814152244, 302563279622819916662, 792109964870135034363, 2073748319114098993735, 5429097556300259429541, 14213467788593013996402, 37211149309391731434874, 97419660384008637002948, 255047178820688210851993, 667720543012566293886157, 1748111730042835739026835, 4576609098681200002743216, 11981704252958500314516800, 31368480601764208803867602, 82123690570782813368071671, 215002495418099348932048339, 562883600839780954870934337, 1473647910496245930925394862, 3858059323603903613112366982, 10100528418555242348261612192, 26443522593201146090243627029, 69230032572646376797354274137, 181246561326573417167074586543, 474509623368022882665351029100, 1242282251813949531213883110476, 3252337016375846880669990566702, 8514728562375858588031239208179, 22291848193792714113589560474463, 58360815050917154764460173407453, 152790594994454293306018550850282, 400010965946768413611614918783890, 1047242294761159528855993599653756, 2741715901940653744432957698224785, 7177905377815341665040577190600677, 18792000164107756807173197383707851, 49198094977899311140025918218724904, 128802284492746163921154476264752472, 337208757939397600322249573511866762, 882823988189056366607842820047042287, 2311263204325976738955024937988700907, 6050965620127255889823221896252045689, 15841633646616498130963616154661613862, 41473935300611539146599568573977565598, 108580172216532493081166955976673450840, 284266581270686232612100996968673179661, 744219571437069879726607359742258187377, 1948392132719896936392807591059228208359, 5100956826073940348866276149388285344228, 13354478344189707668563517765712556146788, 34962478203841039212596421838365660759782, 91532956261965701958884338877421823497195, 239636390571201808399602380543584288535671
Generating function in Maple syntax:
-(x^7-x^6+7*x^5-25*x^4+37*x^3-25*x^2+8*x-1)/(x^2-3*x+1)/(x-1)^2/(2*x-1)^2
Generating function in latex syntax:
-\frac{x^{7}-x^{6}+7 x^{5}-25 x^{4}+37 x^{3}-25 x^{2}+8 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(-x**7 + x**6 - 7*x**5 + 25*x**4 - 37*x**3 + 25*x**2 - 8*x + 1)/((x - 1)**2*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^2*(2*x-1)^2*F(x)+x^7-x^6+7*x^5-25*x^4+37*x^3-25*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{2} F \! \left(x \right)+x^{7}-x^{6}+7 x^{5}-25 x^{4}+37 x^{3}-25 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,(3/2-1/2*5^(1/2))^(1-n)+(3/2+1/2*5^(1/2))^(1-n)+1/8*(3*n-14)*2^n+n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n}+\frac{\left(3 n -14\right) 2^{n}}{8}+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) = 19
a(5) = 57
a(6) = 162
a(7) = 442
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-n, 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) = 57
a \! \left(6\right) = 162
a \! \left(7\right) = 442
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-n, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18536/
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[16,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[7,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[17,x]+F[7,x]
F[17,x] = F[18,x]+F[19,x]+F[66,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[35,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[18,x]+F[23,x]+F[26,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[23,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[11,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[4,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[26,x]
F[33,x] = F[34,x]
F[34,x] = F[31,x]*F[4,x]
F[35,x] = F[17,x]+F[36,x]
F[36,x] = 2*F[18,x]+F[37,x]+F[52,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[51,x]
F[39,x] = F[18,x]+F[40,x]+F[49,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[44,x]
F[42,x] = F[11,x]+F[43,x]
F[43,x] = F[18,x]+F[23,x]+F[30,x]
F[44,x] = F[39,x]+F[45,x]
F[45,x] = 2*F[18,x]+F[37,x]+F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[14,x]+F[48,x]
F[48,x] = F[46,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[39,x]+F[7,x]
F[51,x] = F[37,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[47,x]+F[54,x]
F[54,x] = F[55,x]+F[64,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[14,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[58,x]
F[61,x] = F[55,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[61,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[54,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[50,x]+F[68,x]
F[68,x] = F[69,x]+F[75,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[13,x]+F[72,x]
F[72,x] = F[69,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[72,x]
F[75,x] = F[76,x]
F[76,x] = F[4,x]*F[68,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_{16}\! \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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{66}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{35}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{37}\! \left(x \right)+F_{52}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{51}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{37}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{51}\! \left(x \right) = F_{37}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{47}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{64}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \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_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{55}\! \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_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{50}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{75}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{13}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{69}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \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_16(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_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_7(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_66(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_35(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_18(x) + F_23(x) + F_26(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_23(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_11(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_26(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_31(x)*F_4(x))
Eq(F_35(x), F_17(x) + F_36(x))
Eq(F_36(x), 2*F_18(x) + F_37(x) + F_52(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_51(x))
Eq(F_39(x), F_18(x) + F_40(x) + F_49(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_44(x))
Eq(F_42(x), F_11(x) + F_43(x))
Eq(F_43(x), F_18(x) + F_23(x) + F_30(x))
Eq(F_44(x), F_39(x) + F_45(x))
Eq(F_45(x), 2*F_18(x) + F_37(x) + F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_14(x) + F_48(x))
Eq(F_48(x), F_46(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_39(x) + F_7(x))
Eq(F_51(x), F_37(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_47(x) + F_54(x))
Eq(F_54(x), F_55(x) + F_64(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_14(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_58(x))
Eq(F_61(x), F_55(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_61(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_54(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_50(x) + F_68(x))
Eq(F_68(x), F_69(x) + F_75(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_13(x) + F_72(x))
Eq(F_72(x), F_69(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_4(x)*F_72(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_4(x)*F_68(x))
Pack JSON:
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Specification JSON:
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