0231_1203_1230_1302_3012
Counting sequence:
1, 1, 2, 6, 19, 56, 155, 413, 1078, 2784, 7152, 18329, 46933, 120174, 307840, 789068, 2024042, 5195834, 13348248, 34318110, 88297149, 227345832, 585782438, 1510380400, 3896981273, 10061284935, 25992783563, 67191965577, 173795326576, 449786098460, 1164694938143, 3017502811652, 7821746285398, 20284842628077, 52631156276661, 136618390360832, 354782517802507, 921709478271702, 2395503955405507, 6228208474380806, 16198896567190727, 42146025308004761, 109690698442115277, 285573751253801730, 743696317077113325, 1937294398087402935, 5047917904541818298, 13156484719689089113, 34298320328675576690, 89434678624253133246, 233256680817448264385, 608488565354422345677, 1587655602058065784771, 4143251476257202951130, 10814418438981011834048, 28231755873005308935359, 73712578954287760508419, 192491098762074379108641, 502737795646495204362129, 1313200201897327048170215, 3430644426668999278749955, 8963401210947824554155339, 23421757395569270197587376, 61208667919018940479516205, 159974447586063851233467629, 418148089589367428427977273, 1093072864889713058239020960, 2857626209833225066985376948, 7471313609335986750180761154, 19535376604434590882495723118, 51083192039637445617832290143, 133586916460035180180801543661, 349363731903129100257960964058, 913730520279217950039211503117, 2389920071077355343606652234253, 6251325980062824324538175049630, 16352460529926467124641089331044, 42777461543856585590371745423441, 111909423395456505068526964324262, 292776984129511797082386803936027, 765992952274428304560039943542598, 2004145186265616099053629181145495, 5243837270864041432918932706571946, 13720942985677827303840248081187194, 35903153742582138640927391037424768, 93949468663987650068513102639774319, 245848972733047523863679967242318596, 643360065529464557957919850986729044, 1683645936680008318605025074340014941, 4406134677288661026957176218105421174, 11531200110157995344998038676852577205, 30178693187656890699247812435463533435, 78983250314173138491231276175417584312, 206717729572862605020529768449880331334, 541038453973027385935896268750223715938, 1416073448114751374234350866946975161847, 3706382591056290716466483707872710951143, 9701103595657555714823065332796273605372, 25392069222210391752479511203545080644162, 66463123925187698081330820676675299415074, 173967764650549244842964261085056635793764
Generating function in Maple syntax:
(x^8-9*x^7+31*x^6-66*x^5+86*x^4-70*x^3+34*x^2-9*x+1)/(x^2-3*x+1)/(3*x^3-5*x^2+4*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{8}-9 x^{7}+31 x^{6}-66 x^{5}+86 x^{4}-70 x^{3}+34 x^{2}-9 x +1}{\left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**8 - 9*x**7 + 31*x**6 - 66*x**5 + 86*x**4 - 70*x**3 + 34*x**2 - 9*x + 1)/((x - 1)**3*(x**2 - 3*x + 1)*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(3*x^3-5*x^2+4*x-1)*(x-1)^3*F(x)-x^8+9*x^7-31*x^6+66*x^5-86*x^4+70*x^3-34*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-x^{8}+9 x^{7}-31 x^{6}+66 x^{5}-86 x^{4}+70 x^{3}-34 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/1350360*(-2945*2^(1/3)*((-51/589*31^(1/2)+I)*3^(1/2)-153/589*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(2/3)+300080+3410*((-6/31*31^(1/2)+I)*3^(1/2)+18/31*I*31^(1/2)-1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3))*(47/8712*((I-9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)+1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/1350360*(-3410*((6/31*31^(1/2)+I)*3^(1/2)+18/31*I*31^(1/2)+1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)+300080+2945*2^(1/3)*((51/589*31^(1/2)+I)*3^(1/2)-153/589*I*31^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(2/3))*(-47/8712*2^(1/3)*((I+9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(2/3)+1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/1350360*((1320*2^(2/3)*31^(1/2)*3^(1/2)+6820*2^(2/3))*(47+9*31^(1/2)*3^(1/2))^(1/3)+300080+(-510*2^(1/3)*3^(1/2)*31^(1/2)+5890*2^(1/3))*(47+9*31^(1/2)*3^(1/2))^(2/3))*(1/4356*2^(1/3)*(9*31^(1/2)*3^(1/2)-47)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/18*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/1350360*(-135036*5^(1/2)+675180)*(3/2-1/2*5^(1/2))^(-n)+1/1350360*(135036*5^(1/2)+675180)*(3/2+1/2*5^(1/2))^(-n)-1/2*n^2+1/2*n-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-2945 \,2^{\frac{1}{3}} \left(\left(-\frac{51 \sqrt{31}}{589}+\mathrm{I}\right) \sqrt{3}-\frac{153 \,\mathrm{I} \sqrt{31}}{589}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+300080+3410 \left(\left(-\frac{6 \sqrt{31}}{31}+\mathrm{I}\right) \sqrt{3}+\frac{18 \,\mathrm{I} \sqrt{31}}{31}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{47 \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{1350360}\\+\\\frac{\left(-3410 \left(\left(\frac{6 \sqrt{31}}{31}+\mathrm{I}\right) \sqrt{3}+\frac{18 \,\mathrm{I} \sqrt{31}}{31}+1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+300080+2945 \,2^{\frac{1}{3}} \left(\left(\frac{51 \sqrt{31}}{589}+\mathrm{I}\right) \sqrt{3}-\frac{153 \,\mathrm{I} \sqrt{31}}{589}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{1350360}\\+\\\frac{\left(\left(1320 \,2^{\frac{2}{3}} \sqrt{31}\, \sqrt{3}+6820 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+300080+\left(-510 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{31}+5890 \,2^{\frac{1}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{1350360}\\+\frac{\left(-135036 \sqrt{5}+675180\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{1350360}+\\\frac{\left(135036 \sqrt{5}+675180\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{1350360}-\frac{n^{2}}{2}+\frac{n}{2}-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) = 56
a(6) = 155
a(7) = 413
a(8) = 1078
a(n+5) = -1/2*n^2+3*a(n)-14*a(n+1)+22*a(n+2)-18*a(n+3)+7*a(n+4)-1/2*n-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) = 19
a \! \left(5\right) = 56
a \! \left(6\right) = 155
a \! \left(7\right) = 413
a \! \left(8\right) = 1078
a \! \left(n +5\right) = -\frac{n^{2}}{2}+3 a \! \left(n \right)-14 a \! \left(n +1\right)+22 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right)-\frac{n}{2}-1, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21042/
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[13,x]*F[4,x]
F[4,x] = F[31,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = F[1,x]+F[10,x]
F[13,x] = x
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[14,x]+F[6,x]
F[18,x] = F[13,x]*F[19,x]
F[19,x] = F[20,x]+F[24,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[23,x]
F[23,x] = F[21,x]+F[6,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[10,x]+F[25,x]
F[28,x] = F[29,x]
F[29,x] = F[13,x]*F[30,x]
F[30,x] = F[21,x]+F[28,x]
F[31,x] = F[32,x]+F[42,x]
F[32,x] = F[33,x]
F[33,x] = F[13,x]*F[34,x]
F[34,x] = F[12,x]+F[35,x]
F[35,x] = F[32,x]+F[36,x]
F[36,x] = F[15,x]+F[37,x]+F[41,x]
F[37,x] = F[13,x]*F[38,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[10,x]
F[40,x] = F[36,x]
F[41,x] = F[13,x]*F[35,x]
F[42,x] = F[15,x]+F[43,x]+F[47,x]
F[43,x] = F[13,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[6,x]
F[46,x] = F[42,x]
F[47,x] = F[13,x]*F[48,x]
F[48,x] = F[49,x]+F[54,x]
F[49,x] = F[32,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[13,x]*F[52,x]
F[52,x] = F[45,x]+F[53,x]
F[53,x] = F[50,x]
F[54,x] = F[36,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[13,x]*F[57,x]
F[57,x] = F[50,x]+F[55,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{31}\! \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_{13}\! \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_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \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_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{42}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)
F_{40}\! \left(x \right) = F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{35}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{6}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{49}\! \left(x \right) = F_{32}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{50}\! \left(x \right)
F_{54}\! \left(x \right) = F_{36}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{13}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{50}\! \left(x \right)+F_{55}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_31(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_13(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), F_1(x) + F_10(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_14(x) + F_6(x))
Eq(F_18(x), F_13(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_24(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_23(x))
Eq(F_23(x), F_21(x) + F_6(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_13(x)*F_27(x))
Eq(F_27(x), F_10(x) + F_25(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_13(x)*F_30(x))
Eq(F_30(x), F_21(x) + F_28(x))
Eq(F_31(x), F_32(x) + F_42(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_13(x)*F_34(x))
Eq(F_34(x), F_12(x) + F_35(x))
Eq(F_35(x), F_32(x) + F_36(x))
Eq(F_36(x), F_15(x) + F_37(x) + F_41(x))
Eq(F_37(x), F_13(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_10(x))
Eq(F_40(x), F_36(x))
Eq(F_41(x), F_13(x)*F_35(x))
Eq(F_42(x), F_15(x) + F_43(x) + F_47(x))
Eq(F_43(x), F_13(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_6(x))
Eq(F_46(x), F_42(x))
Eq(F_47(x), F_13(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_54(x))
Eq(F_49(x), F_32(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_13(x)*F_52(x))
Eq(F_52(x), F_45(x) + F_53(x))
Eq(F_53(x), F_50(x))
Eq(F_54(x), F_36(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_13(x)*F_57(x))
Eq(F_57(x), F_50(x) + F_55(x))
Pack JSON:
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Specification JSON:
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