0123_0132_0231_2031_3012
Counting sequence:
1, 1, 2, 6, 19, 57, 163, 454, 1250, 3427, 9381, 25656, 70097, 191291, 521334, 1418872, 3856403, 10467799, 28378595, 76845752, 207860886, 561666461, 1516225717, 4089343726, 11019718449, 29671218837, 79830248218, 214627050794, 576637170915, 1548235739477, 4154337615891, 11140628538666, 29858834726362, 79983884392311, 214145557983989, 573062999412452, 1532821330283233, 4098124451679439, 10951944515457710, 29256163106565852, 78121514349160771, 208524834563814771, 556397383666065907, 1484084044782840796, 3957160541404057454, 10547898223245687217, 28106730269060481573, 74872380362301846234, 199390478012215859105, 530839167479092062377, 1412867298644928535026, 3759443437310551848526, 10000764865631433738227, 26597076007763011261681, 70717835849850679025219, 187984324770189000874382, 499591445453723169822738, 1327431039341601072231179, 3526268448829934891582725, 9365433608174147573041552, 24868643502511484894431281, 66022270978791294340424931, 175244880545336382750502822, 465070729912208820551921216, 1233995675844788994204148659, 3273643038327663941316386447, 8683085294601221252213745603, 23027341671159549303475233024, 61057974340465056700194408070, 161871956389314961434905015877, 429074985323130219475929304661, 1137178896027947531969311916486, 3013422301608677489489884884817, 7984143908894108440697549091901, 21151216526513753232136503067274, 56025071074871443950115975842162, 148378485809333351301889731039043, 392918288282602635312184121344269, 1040345595715663858020877038686515, 2754218246966482823552459307791602, 7290639172812876963656323418294666, 19296589606257331235673966234908191, 51067470622685573897118118616346837, 135131954857193578748680564316146236, 357538450758351270074171555108298689, 945887435250834366356577868635441527, 2502125911683618125918545372901982558, 6618092432035444767285264448755776164, 17502955724439524146672883247317294435, 46285585629098126829054084916366781227, 122387429486283045838715835097343225171, 323582776912220580025452111539172853796, 855445495174359212754490109415145148382, 2261301416300329044450208355227662181721, 5977017282871063360715855761937070473029, 15796878312056695371844331209069836997042, 41746442763386090317138333895062074846273, 110313797428618943932387286286999002896209, 291476166763935778976152176368793145711330, 770087007137066137131638581217072454628502, 2034420550227430827329342931072642067620051
Generating function in Maple syntax:
-(2*x-1)*(x^3-2*x^2+3*x-1)*(x^3-5*x^2+4*x-1)/(x^2-3*x+1)^2/(-1+x)^4
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}-5 x^{2}+4 x -1\right)}{\left(x^{2}-3 x +1\right)^{2} \left(-1+x \right)^{4}}
Generating function in sympy syntax:
(1 - 2*x)*(x**3 - 5*x**2 + 4*x - 1)*(x**3 - 2*x**2 + 3*x - 1)/((x - 1)**4*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)^2*(-1+x)^4*F(x)+(2*x-1)*(x^3-2*x^2+3*x-1)*(x^3-5*x^2+4*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right)^{2} \left(-1+x \right)^{4} F \! \left(x \right)+\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}-5 x^{2}+4 x -1\right) = 0
Explicit closed form in Maple syntax:
1/150*((-45*n-213)*5^(1/2)+105*n+525)*(3/2-1/2*5^(1/2))^(-n)+1/150*((45*n+213)*5^(1/2)+105*n+525)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3-3/2*n^2+13/3*n-6
Explicit closed form in latex syntax:
\frac{\left(\left(-45 n -213\right) \sqrt{5}+105 n +525\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{150}+\frac{\left(\left(45 n +213\right) \sqrt{5}+105 n +525\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{150}+\frac{n^{3}}{6}-\frac{3 n^{2}}{2}+\frac{13 n}{3}-6
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 57
a(6) = 163
a(7) = 454
a(n+4) = -a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3)+1/6*n*(n+2)*(n-5), 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) = 163
a \! \left(7\right) = 454
a \! \left(n +4\right) = -a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right)+\frac{n \left(n +2\right) \left(n -5\right)}{6}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16161/
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[17,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[14,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]
F[15,x] = F[4,x]
F[16,x] = F[12,x]
F[17,x] = F[18,x]+F[32,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[21,x]*F[4,x]
F[24,x] = F[18,x]+F[25,x]
F[25,x] = F[26,x]+F[27,x]+F[31,x]
F[26,x] = 0
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[22,x]
F[30,x] = F[25,x]
F[31,x] = F[24,x]*F[4,x]
F[32,x] = F[26,x]+F[33,x]+F[57,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[40,x]
F[35,x] = F[36,x]+F[7,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[4,x]
F[39,x] = F[37,x]
F[40,x] = F[32,x]+F[41,x]
F[41,x] = 2*F[26,x]+F[42,x]+F[46,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[36,x]
F[45,x] = F[41,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[52,x]
F[48,x] = F[26,x]+F[49,x]+F[51,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[38,x]+F[47,x]
F[51,x] = F[18,x]*F[4,x]
F[52,x] = 2*F[26,x]+F[46,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[39,x]
F[56,x] = F[52,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[18,x]+F[48,x]
F[60,x] = F[61,x]+F[76,x]
F[61,x] = F[26,x]+F[27,x]+F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[70,x]
F[64,x] = F[18,x]+F[65,x]
F[65,x] = F[26,x]+F[51,x]+F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[69,x]
F[68,x] = F[4,x]
F[69,x] = F[65,x]
F[70,x] = F[61,x]+F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[65,x]
F[75,x] = F[71,x]
F[76,x] = 2*F[26,x]+F[53,x]+F[77,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]+F[80,x]
F[79,x] = F[48,x]
F[80,x] = F[76,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_{17}\! \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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
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_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{22}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)+F_{57}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{7}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \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_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{52}\! \left(x \right)
F_{48}\! \left(x \right) = F_{26}\! \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_{38}\! \left(x \right)+F_{47}\! \left(x \right)
F_{51}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{46}\! \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_{39}\! \left(x \right)
F_{56}\! \left(x \right) = F_{52}\! \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_{18}\! \left(x \right)+F_{48}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{76}\! \left(x \right)
F_{61}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{62}\! \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_{70}\! \left(x \right)
F_{64}\! \left(x \right) = F_{18}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{26}\! \left(x \right)+F_{51}\! \left(x \right)+F_{66}\! \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_{69}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right)
F_{69}\! \left(x \right) = F_{65}\! \left(x \right)
F_{70}\! \left(x \right) = F_{61}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{65}\! \left(x \right)
F_{75}\! \left(x \right) = F_{71}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{53}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{80}\! \left(x \right)
F_{79}\! \left(x \right) = F_{48}\! \left(x \right)
F_{80}\! \left(x \right) = F_{76}\! \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_17(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_14(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_12(x))
Eq(F_17(x), F_18(x) + F_32(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_21(x)*F_4(x))
Eq(F_24(x), F_18(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_31(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_22(x))
Eq(F_30(x), F_25(x))
Eq(F_31(x), F_24(x)*F_4(x))
Eq(F_32(x), F_26(x) + F_33(x) + F_57(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_40(x))
Eq(F_35(x), F_36(x) + F_7(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_4(x))
Eq(F_39(x), F_37(x))
Eq(F_40(x), F_32(x) + F_41(x))
Eq(F_41(x), 2*F_26(x) + F_42(x) + F_46(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_36(x))
Eq(F_45(x), F_41(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_52(x))
Eq(F_48(x), F_26(x) + F_49(x) + F_51(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_38(x) + F_47(x))
Eq(F_51(x), F_18(x)*F_4(x))
Eq(F_52(x), 2*F_26(x) + F_46(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_39(x))
Eq(F_56(x), F_52(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_18(x) + F_48(x))
Eq(F_60(x), F_61(x) + F_76(x))
Eq(F_61(x), F_26(x) + F_27(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_70(x))
Eq(F_64(x), F_18(x) + F_65(x))
Eq(F_65(x), F_26(x) + F_51(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_69(x))
Eq(F_68(x), F_4(x))
Eq(F_69(x), F_65(x))
Eq(F_70(x), F_61(x) + F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_65(x))
Eq(F_75(x), F_71(x))
Eq(F_76(x), 2*F_26(x) + F_53(x) + F_77(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_80(x))
Eq(F_79(x), F_48(x))
Eq(F_80(x), F_76(x))
Pack JSON:
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Specification JSON:
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