0123_0132_0213_1032_2130

Counting sequence:
1, 1, 2, 6, 19, 55, 148, 382, 963, 2391, 5880, 14370, 34975, 84895, 205696, 497794, 1203723, 2909187, 7028484, 16976490, 40998187, 98999923, 239041816, 577154398, 1393465239, 3364270347, 8122306032, 19609367982, 47341827667, 114294294559, 275932473700, 666162570118, 1608262999011, 3882697281375, 9373671660072, 22630063413066, 54633835396063, 131897793926599, 318429519880528, 768756990040330, 1855943752945131, 4480644905267211, 10817234225800116, 26115114428524626, 63047464816827115, 152210046867813787, 367467563092067368, 887145180397196134, 2141757935771319927, 5170661071169943891, 12483080109226175904, 30136821339967371798, 72756722870620963795, 175650267213014419783, 424057257509914968052, 1023764782577914640974, 2471586823224079699779, 5966938429929479775399, 14405463684544780435224, 34777865801384187565362, 83961195291140043670111, 202700256389856309929263, 489361708080871586656480, 1181423672567810441393746, 2852209053242722350723339, 6885841779095695982271315, 16623892611502785035976228, 40133627002212377614364922, 96891146616107322545557483, 233915920234717916546472451, 564722987086013831760346360, 1363361894407507150030001710, 3291446775902260377905030295, 7946255446214021721887579355, 19183957668333529883812386576, 46314170782886301367692067134, 111812299234114578559508433043, 269938769251129124305200560047, 651689837736394938928713092164, 1573318444723954779739921910230, 3798326727184362387744655617507, 9169971899092773222142627015983, 22138270525370060388279402225096, 53446512949833139221864317912538, 129031296425036735611420417072159, 311509105799907252447280417525207, 752049508024852279287968896612912, 1815608121849613491807781120709722, 4383265751724081982470081692481387, 10582139625297781857099057970080219, 25547545002319652816585861651498580, 61677229629937099010539558756341858, 148902004262193869477851420666303531, 359481238154324868126697619074334635, 867864480570843654531888319302479752, 2095210199296012256151571137152186806, 5058284879162868294596769133567252983, 12211779957621749052067944823719985059, 29481844794406366733217232740400915008, 71175469546434483059709819683348799270, 171832783887275333728328855445319189651

Generating function in Maple syntax:
(2*x^7+2*x^6-2*x^4-4*x^3-x^2+3*x-1)/(x-1)/(x^2+x-1)/(x^2+2*x-1)

Generating function in latex syntax:
\frac{2 x^{7}+2 x^{6}-2 x^{4}-4 x^{3}-x^{2}+3 x -1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right)}

Generating function in sympy syntax:
(2*x**7 + 2*x**6 - 2*x**4 - 4*x**3 - x**2 + 3*x - 1)/((x - 1)*(x**2 + x - 1)*(x**2 + 2*x - 1))

Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^2+x-1)*(x^2+2*x-1)*F(x)+2*x^7+2*x^6-2*x^4-4*x^3-x^2+3*x-1 = 0

Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+2 x^{7}+2 x^{6}-2 x^{4}-4 x^{3}-x^{2}+3 x -1 = 0

Explicit closed form in Maple syntax:
2*piecewise(n = 0,5,n = 1,-1,n = 2,1,0)+1/2+4/5*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-4/5*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+4*(2^(1/2)-1)^(-n)*2^(1/2)-4*(-1-2^(1/2))^(-n)*2^(1/2)-19/4*(2^(1/2)-1)^(-n)-19/4*(-1-2^(1/2))^(-n)

Explicit closed form in latex syntax:
2 \left(\left\{\begin{array}{cc}5 & n =0 \\ -1 & n =1 \\ 1 & n =2 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)+\frac{1}{2}+\frac{4 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}-\frac{4 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+4 \left(\sqrt{2}-1\right)^{-n} \sqrt{2}-4 \left(-1-\sqrt{2}\right)^{-n} \sqrt{2}-\frac{19 \left(\sqrt{2}-1\right)^{-n}}{4}-\frac{19 \left(-1-\sqrt{2}\right)^{-n}}{4}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 148
a(7) = 382
a(n+1) = -1/3*a(n)+a(n+3)-1/3*a(n+4)+1/3, 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) = 148
a \! \left(7\right) = 382
a \! \left(n +1\right) = -\frac{a \! \left(n \right)}{3}+a \! \left(n +3\right)-\frac{a \! \left(n +4\right)}{3}+\frac{1}{3}, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15981/
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[14,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[4,x]*F[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[62,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[27,x]
F[19,x] = F[20,x]+F[22,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[4,x]
F[22,x] = F[16,x]+F[23,x]+F[26,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]
F[25,x] = F[4,x]
F[26,x] = F[11,x]*F[4,x]
F[27,x] = F[28,x]+F[61,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[33,x]
F[31,x] = F[20,x]+F[32,x]
F[32,x] = F[23,x]
F[33,x] = F[28,x]+F[34,x]
F[34,x] = 2*F[16,x]+F[35,x]+F[44,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[39,x]
F[37,x] = F[38,x]
F[38,x] = F[25,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[38,x]
F[43,x] = F[40,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[47,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[23,x]
F[51,x] = F[46,x]+F[52,x]
F[52,x] = 2*F[16,x]+F[44,x]+F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[38,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[38,x]
F[60,x] = F[57,x]
F[61,x] = 2*F[16,x]+F[35,x]+F[44,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[73,x]
F[64,x] = F[2,x]+F[65,x]
F[65,x] = F[16,x]+F[66,x]+F[72,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[70,x]
F[68,x] = F[4,x]+F[69,x]
F[69,x] = F[13,x]+F[16,x]+F[23,x]
F[70,x] = F[46,x]+F[71,x]
F[71,x] = 2*F[16,x]+F[44,x]+F[53,x]
F[72,x] = F[2,x]*F[4,x]
F[73,x] = F[15,x]+F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[15,x]*F[4,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_{14}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{62}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{27}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \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_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{61}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{33}\! \left(x \right)
F_{31}\! \left(x \right) = F_{20}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{23}\! \left(x \right)
F_{33}\! \left(x \right) = F_{28}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \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_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \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_{47}\! \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_{23}\! \left(x \right)
F_{51}\! \left(x \right) = F_{46}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{44}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \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_{38}\! \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_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{38}\! \left(x \right)
F_{60}\! \left(x \right) = F_{57}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{73}\! \left(x \right)
F_{64}\! \left(x \right) = F_{2}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{16}\! \left(x \right)+F_{66}\! \left(x \right)+F_{72}\! \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_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{13}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{70}\! \left(x \right) = F_{46}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{44}\! \left(x \right)+F_{53}\! \left(x \right)
F_{72}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{73}\! \left(x \right) = F_{15}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \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_14(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_4(x)*F_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_62(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_27(x))
Eq(F_19(x), F_20(x) + F_22(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_4(x))
Eq(F_22(x), F_16(x) + F_23(x) + F_26(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_4(x))
Eq(F_26(x), F_11(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_61(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_33(x))
Eq(F_31(x), F_20(x) + F_32(x))
Eq(F_32(x), F_23(x))
Eq(F_33(x), F_28(x) + F_34(x))
Eq(F_34(x), 2*F_16(x) + F_35(x) + F_44(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_39(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_25(x)*F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_38(x))
Eq(F_43(x), F_40(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_47(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_23(x))
Eq(F_51(x), F_46(x) + F_52(x))
Eq(F_52(x), 2*F_16(x) + F_44(x) + F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_38(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) + F_60(x))
Eq(F_59(x), F_38(x))
Eq(F_60(x), F_57(x))
Eq(F_61(x), 2*F_16(x) + F_35(x) + F_44(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_73(x))
Eq(F_64(x), F_2(x) + F_65(x))
Eq(F_65(x), F_16(x) + F_66(x) + F_72(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_70(x))
Eq(F_68(x), F_4(x) + F_69(x))
Eq(F_69(x), F_13(x) + F_16(x) + F_23(x))
Eq(F_70(x), F_46(x) + F_71(x))
Eq(F_71(x), 2*F_16(x) + F_44(x) + F_53(x))
Eq(F_72(x), F_2(x)*F_4(x))
Eq(F_73(x), F_15(x) + F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_15(x)*F_4(x))
Pack JSON:
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Specification JSON:
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