0213_0321_1032_1302_2130
Counting sequence:
1, 1, 2, 6, 19, 56, 156, 423, 1136, 3042, 8135, 21726, 57940, 154301, 410394, 1090260, 2893367, 7671162, 20320634, 53785137, 142253752, 375981578, 993101133, 2621592054, 6916721582, 18239638191, 48076032920, 126663632448, 333579548791, 878179411784, 2311078371798, 6080006508363, 15990459139694, 42043047888276, 110512654917605, 290416967505846, 763011629135476, 2004223104013631, 5263477663603818, 13820284033632810, 36281386673481701, 95231014139480916, 249922369615205054, 655795296132328425, 1720568169593342770, 4513563950373711206, 11838996982652605923, 31049960465887997904, 81425615995780911740, 213509766406387821639, 559801141264178944736, 1467613289866977729042, 3847278604710886403455, 10084676241480455717894, 26432524504161403040604, 69276445851775967865765, 181553496603745069661906, 475770358419205212102708, 1246710064472278248436703, 3266708256630266112771210, 8559204906236265912128002, 22425182614641555060169505, 58751541808749673642043400, 153916228135490683644581922, 403211117995637946847542965, 1056244697436853169349291894, 2766823739638713416561973086, 7247433651041051148194486031, 18983384619868827059998452592, 49722216482265533386504369856, 130231334278491543562017387495, 341090305010116823471972126904, 893331796942771289295193172054, 2339625565651999174704170624787, 6127321294364596122290373753062, 16046718144250995177930237196284, 42023631913369770308059898536141, 110050802744791919066090433556382, 288194422522261113644067358893876, 754694320207288054972009276548567, 1976287604081314759885794146073482, 5175152417640785273872787566124514, 13551595587504888614730750693772277, 35485615669453760662216622084782980, 92919998802669063122079078344936134, 243310741458896249904169222663926641, 637101875520067262915914687289423298, 1668215923429219747395256069558233606, 4368090880510039395449593783729645043, 11437400419269259789890538472166458376, 29947423368242690922726416774252590940, 78413038100624152266160447250877076167, 205311830743071452337530930443714650736, 537572110263365106999690874364630702690, 1407526930784411063229310992661809983223, 3685310543793867409078354668641518256814, 9649148962130577203487132364140549677060, 25263971372423573311061304902222652862221, 66147289551888046950173273196970629841322, 173189052505703796991417303766882924618868, 453447371960813868058566127579564357448743
Generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x^3-5*x^2+4*x-1)/(x-1)/(x^2-3*x+1)/(3*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
\frac{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}-5 x^{2}+4 x -1\right)}{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(x**3 - 5*x**2 + 4*x - 1)*(x**3 - 2*x**2 + 3*x - 1)/((x - 1)*(x**2 - 3*x + 1)*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2-3*x+1)*(3*x^3-5*x^2+4*x-1)*F(x)-(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 -1\right) \left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \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:
piecewise(n = 0,1,1/675180*(4805*2^(1/3)*((I-45/961*31^(1/2))*3^(1/2)-135/961*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(2/3)+150040+1705*2^(2/3)*((I+27/31*31^(1/2))*3^(1/2)-81/31*I*31^(1/2)-1)*(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)+1/675180*(-1705*2^(2/3)*((I-27/31*31^(1/2))*3^(1/2)-81/31*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(1/3)+150040-4805*2^(1/3)*((I+45/961*31^(1/2))*3^(1/2)-135/961*I*31^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(2/3))*(-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)+1/675180*((-2970*3^(1/2)*31^(1/2)*2^(2/3)+3410*2^(2/3))*(47+9*31^(1/2)*3^(1/2))^(1/3)+150040+(450*31^(1/2)*3^(1/2)*2^(1/3)-9610*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)+1/675180*(67518*5^(1/2)+337590)*(3/2-1/2*5^(1/2))^(-n)-1+1/675180*(-67518*5^(1/2)+337590)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(4805 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{45 \sqrt{31}}{961}\right) \sqrt{3}-\frac{135 \,\mathrm{I} \sqrt{31}}{961}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+150040+1705 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{27 \sqrt{31}}{31}\right) \sqrt{3}-\frac{81 \,\mathrm{I} \sqrt{31}}{31}-1\right) \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}}{675180}\\+\\\frac{\left(-1705 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{27 \sqrt{31}}{31}\right) \sqrt{3}-\frac{81 \,\mathrm{I} \sqrt{31}}{31}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+150040-4805 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{45 \sqrt{31}}{961}\right) \sqrt{3}-\frac{135 \,\mathrm{I} \sqrt{31}}{961}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{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}}{675180}\\+\\\frac{\left(\left(-2970 \sqrt{3}\, \sqrt{31}\, 2^{\frac{2}{3}}+3410 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+150040+\left(450 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}-9610 \,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}}{675180}\\+\frac{\left(67518 \sqrt{5}+337590\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{675180}-1+\\\frac{\left(-67518 \sqrt{5}+337590\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{675180} & \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) = 156
a(n+5) = 3*a(n)-14*a(n+1)+22*a(n+2)-18*a(n+3)+7*a(n+4)-1, n >= 7
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) = 156
a \! \left(n +5\right) = 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)-1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20494/
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[17,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[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] = x
F[13,x] = F[1,x]+F[10,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[9,x]
F[17,x] = F[18,x]+F[32,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[5,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[21,x]+F[25,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[18,x]+F[26,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[25,x]
F[32,x] = F[33,x]+F[34,x]+F[47,x]
F[33,x] = 0
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[40,x]
F[36,x] = F[10,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[36,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[33,x]+F[34,x]+F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[18,x]+F[41,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[40,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[17,x]+F[49,x]
F[49,x] = F[50,x]+F[57,x]
F[50,x] = F[27,x]+F[33,x]+F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[36,x]+F[53,x]
F[53,x] = F[26,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[53,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[49,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_{17}\! \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_{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) = x
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \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_{9}\! \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_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{5}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{47}\! \left(x \right)
F_{33}\! \left(x \right) = 0
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{40}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{17}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{57}\! \left(x \right)
F_{50}\! \left(x \right) = F_{27}\! \left(x \right)+F_{33}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{36}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{26}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{53}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{49}\! \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_17(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_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), x)
Eq(F_13(x), F_1(x) + F_10(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_9(x))
Eq(F_17(x), F_18(x) + F_32(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_5(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_21(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_18(x) + F_26(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_25(x))
Eq(F_32(x), F_33(x) + F_34(x) + F_47(x))
Eq(F_33(x), 0)
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_40(x))
Eq(F_36(x), F_10(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_36(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_33(x) + F_34(x) + F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_18(x) + F_41(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_40(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_17(x) + F_49(x))
Eq(F_49(x), F_50(x) + F_57(x))
Eq(F_50(x), F_27(x) + F_33(x) + F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_36(x) + F_53(x))
Eq(F_53(x), F_26(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_53(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_49(x))
Pack JSON:
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Specification JSON:
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