0132_0321_1032_1230_3120
Counting sequence:
1, 1, 2, 6, 19, 51, 122, 273, 587, 1231, 2540, 5185, 10509, 21199, 42630, 85553, 171471, 343391, 687328, 1375313, 2751409, 5503743, 11008570, 22018401, 44038259, 88078191, 176158292, 352318753, 704639957, 1409282671, 2818568430, 5637140305, 11274284439, 22548573119, 45097150920, 90194306993, 180388619641, 360777245471, 721554497698, 1443109002753, 2886218013499, 5772436035663, 11544872080700, 23089744171521, 46179488353949, 92358976719631, 184717953451862, 369435906917233, 738871813848927, 1477743627713311, 2955487255443120, 5910974510903825, 11821949021826369, 23643898043672639, 47287796087366410, 94575592174755233, 189151184349534211, 378302368699093551, 756604737398213668, 1513209474796455393, 3026418949592940389, 6052837899185911983, 12105675798371856830, 24211351596743748241, 48422703193487532839, 96845406386975103871, 193690812773950247832, 387381625547900537713, 774763251095801119497, 1549526502191602285151, 3099053004383204618610, 6198106008766409287745, 12396212017532818628299, 24792424035065637311759, 49584848070131274681100, 99169696140262549422273, 198339392280525098907181, 396678784561050197879631, 793357569122100395827238, 1586715138244200791725233, 3173430276488401583524079, 6346860552976803167124703, 12693721105953606334328960, 25387442211907212668740561, 50774884423814425337566929, 101549768847628850675222911, 203099537695257701350538202, 406199075390515402701172193, 812398150781030805402443667, 1624796301562061610804990191, 3249592603124123221610086900, 6499185206248246443220284065, 12998370412496492886440682229, 25996740824992985772881482479, 51993481649985971545763086990, 103986963299971943091526300113, 207973926599943886183052730551, 415947853199887772366105595711, 831895706399775544732211330408, 1663791412799551089464422804273, 3327582825599102178928845756569
Generating function in Maple syntax:
-(x^7+2*x^6-4*x^5+4*x^4-8*x^3+10*x^2-5*x+1)/(2*x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{x^{7}+2 x^{6}-4 x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1}{\left(2 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-x**7 - 2*x**6 + 4*x**5 - 4*x**4 + 8*x**3 - 10*x**2 + 5*x - 1)/((x - 1)**4*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^4*F(x)+x^7+2*x^6-4*x^5+4*x^4-8*x^3+10*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{7}+2 x^{6}-4 x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,7+21/8*2^n-53/6*n+n^2-1/6*n^3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 7+\frac{21 \,2^{n}}{8}-\frac{53 n}{6}+n^{2}-\frac{n^{3}}{6} & \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) = 51
a(6) = 122
a(7) = 273
a(n+1) = 1/6*n^3-3/2*n^2+2*a(n)+31/3*n-15, 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) = 51
a \! \left(6\right) = 122
a \! \left(7\right) = 273
a \! \left(n +1\right) = \frac{n^{3}}{6}-\frac{3 n^{2}}{2}+2 a \! \left(n \right)+\frac{31 n}{3}-15, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19370/
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[24,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[12,x]
F[17,x] = F[18,x]+F[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[10,x]*F[12,x]
F[23,x] = F[12,x]*F[13,x]
F[24,x] = F[25,x]+F[38,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[16,x]
F[31,x] = F[25,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[12,x]
F[36,x] = F[37,x]
F[37,x] = F[33,x]
F[38,x] = F[18,x]+F[39,x]+F[61,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[48,x]
F[41,x] = F[10,x]+F[42,x]
F[42,x] = F[18,x]+F[43,x]+F[46,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[18,x]+F[22,x]+F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[12,x]+F[45,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[18,x]+F[39,x]+F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[25,x]+F[49,x]
F[52,x] = 2*F[18,x]+F[53,x]+F[58,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[21,x]
F[56,x] = F[57,x]
F[57,x] = F[53,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[37,x]+F[60,x]
F[60,x] = 2*F[18,x]+F[53,x]+F[58,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[51,x]+F[63,x]
F[63,x] = F[64,x]+F[73,x]
F[64,x] = F[18,x]+F[65,x]+F[71,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]+F[70,x]
F[67,x] = F[12,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = x^2
F[70,x] = F[37,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[25,x]+F[37,x]
F[73,x] = F[18,x]+F[74,x]+F[80,x]+F[82,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[76,x]+F[79,x]
F[76,x] = F[21,x]+F[77,x]
F[77,x] = F[78,x]
F[78,x] = F[12,x]*F[45,x]
F[79,x] = F[57,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[49,x]+F[57,x]
F[82,x] = F[12,x]*F[63,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_{24}\! \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_{17}\! \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_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{38}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{61}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{48}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \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_{12}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{25}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \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_{21}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{37}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{51}\! \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_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{71}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{70}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = x^{2}
F_{70}\! \left(x \right) = F_{37}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{25}\! \left(x \right)+F_{37}\! \left(x \right)
F_{73}\! \left(x \right) = F_{18}\! \left(x \right)+F_{74}\! \left(x \right)+F_{80}\! \left(x \right)+F_{82}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{79}\! \left(x \right)
F_{76}\! \left(x \right) = F_{21}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{79}\! \left(x \right) = F_{57}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{49}\! \left(x \right)+F_{57}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \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_24(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_12(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_10(x)*F_12(x))
Eq(F_23(x), F_12(x)*F_13(x))
Eq(F_24(x), F_25(x) + F_38(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_16(x))
Eq(F_31(x), F_25(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_12(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_33(x))
Eq(F_38(x), F_18(x) + F_39(x) + F_61(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_48(x))
Eq(F_41(x), F_10(x) + F_42(x))
Eq(F_42(x), F_18(x) + F_43(x) + F_46(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_18(x) + F_22(x) + F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_12(x) + F_45(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_18(x) + F_39(x) + F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_25(x) + F_49(x))
Eq(F_52(x), 2*F_18(x) + F_53(x) + F_58(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_21(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_37(x) + F_60(x))
Eq(F_60(x), 2*F_18(x) + F_53(x) + F_58(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_51(x) + F_63(x))
Eq(F_63(x), F_64(x) + F_73(x))
Eq(F_64(x), F_18(x) + F_65(x) + F_71(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_70(x))
Eq(F_67(x), F_12(x) + F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), x**2)
Eq(F_70(x), F_37(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_25(x) + F_37(x))
Eq(F_73(x), F_18(x) + F_74(x) + F_80(x) + F_82(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_79(x))
Eq(F_76(x), F_21(x) + F_77(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_12(x)*F_45(x))
Eq(F_79(x), F_57(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_49(x) + F_57(x))
Eq(F_82(x), F_12(x)*F_63(x))
Pack JSON:
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Specification JSON:
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