0123_1302_1320_2031_3102
Counting sequence:
1, 1, 2, 6, 19, 51, 120, 264, 557, 1145, 2315, 4627, 9169, 18045, 35305, 68713, 133096, 256670, 492959, 943199, 1798362, 3417832, 6476446, 12238878, 23070930, 43391068, 81439240, 152561532, 285303977, 532708413, 993235126, 1849489766, 3439860347, 6390956363, 11862330985, 21998472489, 40763265695, 75479986655, 139672392759, 258304563183, 477442107166, 882061058400, 1628868381317, 3006775601661, 5548304649992, 10234788606274, 18874279652836, 34797431202116, 64138747288928, 118195350531254, 217769034245982, 401158480954754, 738869036132767, 1360684338191367, 2505485396397092, 4612925950128020, 8492121840534537, 15632037809011165, 28772573099466007, 52955272879507983, 97456416149052109, 179343387359123521, 330017291450164021, 607249045461771269, 1117321057517144340, 2055758456068627874, 3782255899503385787, 6958506538644792235, 12801737029258738838, 23551077120495188444, 43325422538050089306, 79701414644555532218, 146616000625036739630, 269705407055874729808, 496125589380713272580, 912615749819726282712, 1678723794009362975733, 3087919878115276547921, 5680005686936913184066, 10447873785687922563794, 19217807931421402667095, 35348981717144620247407, 65020065522886712430869, 119595712073827700490629, 219979278000363676382299, 404618851382098425144931, 744232835819452419677283, 1368894855437875840799467, 2517851207217966627582538, 4631150333356990520423332, 8518177155519678938652481, 15667638430596982415207721, 28817718613643259274548356, 53004766368706094225437302, 97492140216496751902578048, 179317925852051549845291200, 329820233515153746251832508, 606639136616659187427691882, 1115791753017716807786491130, 2052274771441187384417586494, 3774744339548917360662090155
Generating function in Maple syntax:
-(x^11+3*x^10+4*x^9-4*x^8-13*x^7-x^6+6*x^5+6*x^4-8*x^2+5*x-1)/(x^3+x^2+x-1)/(x^2+x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{x^{11}+3 x^{10}+4 x^{9}-4 x^{8}-13 x^{7}-x^{6}+6 x^{5}+6 x^{4}-8 x^{2}+5 x -1}{\left(x^{3}+x^{2}+x -1\right) \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**11 - 3*x**10 - 4*x**9 + 4*x**8 + 13*x**7 + x**6 - 6*x**5 - 6*x**4 + 8*x**2 - 5*x + 1)/((x - 1)**3*(x**2 + x - 1)**2*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+x^2+x-1)*(x^2+x-1)^2*(x-1)^3*F(x)+x^11+3*x^10+4*x^9-4*x^8-13*x^7-x^6+6*x^5+6*x^4-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+x^{2}+x -1\right) \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+x^{11}+3 x^{10}+4 x^{9}-4 x^{8}-13 x^{7}-x^{6}+6 x^{5}+6 x^{4}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/13200*(((19525*I-4875*11^(1/2))*3^(1/2)-14625*I*11^(1/2)+19525)*(17+3*11^(1/2)*3^(1/2))^(1/3)+57200+((37675*I+6075*11^(1/2))*3^(1/2)-18225*I*11^(1/2)-37675)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/13200*(((-37675*I+6075*11^(1/2))*3^(1/2)+18225*I*11^(1/2)-37675)*(17+3*11^(1/2)*3^(1/2))^(2/3)+57200+((-19525*I-4875*11^(1/2))*3^(1/2)+14625*I*11^(1/2)+19525)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/13200*((-12150*11^(1/2)*3^(1/2)+75350)*(17+3*11^(1/2)*3^(1/2))^(2/3)+9750*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-39050*(17+3*11^(1/2)*3^(1/2))^(1/3)+57200)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/13200*((6600*n+12936)*5^(1/2)-14520*n-33000)*(-1/2*5^(1/2)-1/2)^(-n)+1/13200*((-6600*n-12936)*5^(1/2)-14520*n-33000)*(1/2*5^(1/2)-1/2)^(-n)-1/2*n^2-6*n-4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(19525 \,\mathrm{I}-4875 \sqrt{11}\right) \sqrt{3}-14625 \,\mathrm{I} \sqrt{11}+19525\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+57200+\left(\left(37675 \,\mathrm{I}+6075 \sqrt{11}\right) \sqrt{3}-18225 \,\mathrm{I} \sqrt{11}-37675\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{13200}\\+\\\frac{\left(\left(\left(-37675 \,\mathrm{I}+6075 \sqrt{11}\right) \sqrt{3}+18225 \,\mathrm{I} \sqrt{11}-37675\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+57200+\left(\left(-19525 \,\mathrm{I}-4875 \sqrt{11}\right) \sqrt{3}+14625 \,\mathrm{I} \sqrt{11}+19525\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{13200}\\+\\\frac{\left(\left(-12150 \sqrt{11}\, \sqrt{3}+75350\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+9750 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-39050 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+57200\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{13200}\\+\frac{\left(\left(6600 n +12936\right) \sqrt{5}-14520 n -33000\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{13200}+\\\frac{\left(\left(-6600 n -12936\right) \sqrt{5}-14520 n -33000\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{13200}-\frac{n^{2}}{2}-6 n -4 & \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) = 120
a(7) = 264
a(8) = 557
a(9) = 1145
a(10) = 2315
a(11) = 4627
a(n+4) = 1/4*n^2+1/4*a(n)+3/4*a(n+1)+1/2*a(n+2)-1/2*a(n+3)+3/4*a(n+6)-1/4*a(n+7)+2*n-11/2, n >= 12
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) = 120
a \! \left(7\right) = 264
a \! \left(8\right) = 557
a \! \left(9\right) = 1145
a \! \left(10\right) = 2315
a \! \left(11\right) = 4627
a \! \left(n +4\right) = \frac{n^{2}}{4}+\frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(n +2\right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}+2 n -\frac{11}{2}, \quad n \geq 12
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18060/
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[27,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[21,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[14,x]
F[21,x] = F[19,x]+F[22,x]+F[23,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[11,x]
F[26,x] = F[21,x]
F[27,x] = F[28,x]+F[40,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[1,x]+F[14,x]
F[32,x] = F[28,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]
F[36,x] = F[28,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[28,x]
F[40,x] = F[22,x]+F[41,x]+F[59,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[49,x]
F[43,x] = F[44,x]+F[7,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[25,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[45,x]
F[49,x] = F[40,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[22,x]+F[38,x]+F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[11,x]+F[48,x]
F[57,x] = F[53,x]+F[58,x]
F[58,x] = F[51,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[28,x]+F[53,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_{27}\! \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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{21}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \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_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = 0
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_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{40}\! \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_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{28}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{28}\! \left(x \right)
F_{40}\! \left(x \right) = F_{22}\! \left(x \right)+F_{41}\! \left(x \right)+F_{59}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{7}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{25}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{45}\! \left(x \right)
F_{49}\! \left(x \right) = F_{40}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{22}\! \left(x \right)+F_{38}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{11}\! \left(x \right)+F_{48}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{51}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{28}\! \left(x \right)+F_{53}\! \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_27(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_14(x))
Eq(F_21(x), F_19(x) + F_22(x) + F_23(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_11(x))
Eq(F_26(x), F_21(x))
Eq(F_27(x), F_28(x) + F_40(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_32(x))
Eq(F_31(x), F_1(x) + F_14(x))
Eq(F_32(x), F_28(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_28(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_28(x))
Eq(F_40(x), F_22(x) + F_41(x) + F_59(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_49(x))
Eq(F_43(x), F_44(x) + F_7(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_25(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_45(x))
Eq(F_49(x), F_40(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_22(x) + F_38(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_11(x) + F_48(x))
Eq(F_57(x), F_53(x) + F_58(x))
Eq(F_58(x), F_51(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_28(x) + F_53(x))
Pack JSON:
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Specification JSON:
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