0123_0231_1032_2031_2130
Counting sequence:
1, 1, 2, 6, 19, 54, 144, 372, 941, 2342, 5755, 14001, 33794, 81054, 193414, 459607, 1088406, 2570137, 6054645, 14234946, 33411444, 78310604, 183326909, 428737042, 1001800071, 2339117357, 5458217610, 12729675466, 29674623094, 69148565079, 161077782754, 375114157677, 873342105561, 2032889050002, 4731123628400, 11008987954436, 25613711783869, 59586495325854, 138605474873699, 322385994772137, 749791227932626, 1743722565202822, 4055002026666390, 9429413761828231, 21926082965958894, 50982669738094929, 118541749718611869, 275618977994437922, 640822074981851516, 1489901960070623692, 3463944591033272205, 8053379693521753226, 18723207389732003919, 43528918435511107429, 101197942345241727514, 235267641803736291282, 546952847654839165606, 1271554978699491920487, 2956094436586944278970, 6872261308005509276421, 15976420245519773421569, 37141372989090530450354, 86344620648148122703384, 200729697939604218469092, 466645548967249216127021, 1084830518581971885498582, 2521947449573590545415435, 5862861448169215286785441, 13629589139232713065842594, 31685130321513851991009454, 73659350835406450771402854, 171237934404746572684580727, 398081338669585091075042886, 925429711643957423773563977, 2151368819216026352147574725, 5001337227466904680555414786, 11626721464486929522314210244, 27028894175491627080318478604, 62834647660465169921560635549, 146073017765893671215715570562, 339578935494231273063056740951, 789425985295349591036725360797, 1835194236031435786914629458090, 4266311939733751229163196967706, 9917977865905555259556462495894, 23056513021224246216976433306679, 53599913405482673580417192488594, 124604728307679672005730414533597, 289670943668847091951083774300841, 673403432867762959177708651065042, 1565473429715471073147690748609280, 3639284947364127808060742576963076, 8460312576097991067108562602664669, 19667843584418762318439908838535054, 45722195190699579949339374758954739, 106291220263909500918598930945890457, 247097132563848694456262075645160306, 574431189492627840042335986776381462, 1335390597890497989255899233932167670, 3104406695802891696384920791783807271, 7216870378225916379176565870053281694
Generating function in Maple syntax:
-(x^7+x^6-3*x^5+3*x^4-6*x^3+9*x^2-5*x+1)/(x-1)/(2*x-1)/(x^3-2*x^2+3*x-1)
Generating function in latex syntax:
-\frac{x^{7}+x^{6}-3 x^{5}+3 x^{4}-6 x^{3}+9 x^{2}-5 x +1}{\left(x -1\right) \left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(-x**7 - x**6 + 3*x**5 - 3*x**4 + 6*x**3 - 9*x**2 + 5*x - 1)/((x - 1)*(2*x - 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)*(x^3-2*x^2+3*x-1)*F(x)+x^7+x^6-3*x^5+3*x^4-6*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+x^{7}+x^{6}-3 x^{5}+3 x^{4}-6 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,1/2760*(23*((-9/23*23^(1/2)+I)*3^(1/2)-27/23*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+4600-161*2^(2/3)*((3/161*23^(1/2)+I)*3^(1/2)-9/161*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/2760*(161*2^(2/3)*((-3/161*23^(1/2)+I)*3^(1/2)-9/161*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+4600-23*2^(1/3)*((9/23*23^(1/2)+I)*3^(1/2)-27/23*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/2760*((6*2^(2/3)*23^(1/2)*3^(1/2)-322*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+4600+(18*23^(1/2)*3^(1/2)*2^(1/3)-46*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-15/8*2^n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(23 \left(\left(-\frac{9 \sqrt{23}}{23}+\mathrm{I}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{23}}{23}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+4600-161 \,2^{\frac{2}{3}} \left(\left(\frac{3 \sqrt{23}}{161}+\mathrm{I}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{161}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{2760}\\+\\\frac{\left(161 \,2^{\frac{2}{3}} \left(\left(-\frac{3 \sqrt{23}}{161}+\mathrm{I}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{161}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+4600-23 \,2^{\frac{1}{3}} \left(\left(\frac{9 \sqrt{23}}{23}+\mathrm{I}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{23}}{23}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{2760}\\+\\\frac{\left(\left(6 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}-322 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+4600+\left(18 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}-46 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{2760}\\-\frac{15 \,2^{n}}{8}+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) = 54
a(6) = 144
a(7) = 372
a(n+4) = -2*a(n)+5*a(n+1)-8*a(n+2)+5*a(n+3)+1, 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) = 54
a \! \left(6\right) = 144
a \! \left(7\right) = 372
a \! \left(n +4\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right)+1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17259/
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[20,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[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[15,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[53,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[30,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[1,x]+F[4,x]
F[29,x] = F[15,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[23,x]
F[32,x] = 2*F[22,x]+F[33,x]+F[39,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[4,x]
F[38,x] = F[33,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[52,x]
F[41,x] = F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[19,x]+F[4,x]
F[46,x] = F[42,x]+F[47,x]
F[47,x] = 2*F[22,x]+F[39,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[36,x]
F[51,x] = F[48,x]
F[52,x] = F[47,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[58,x]
F[55,x] = F[2,x]+F[56,x]
F[56,x] = F[22,x]+F[43,x]+F[57,x]
F[57,x] = F[4,x]*F[55,x]
F[58,x] = F[21,x]+F[59,x]
F[59,x] = 2*F[22,x]+F[39,x]+F[48,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_{20}\! \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_{7}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{53}\! \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_{30}\! \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_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right)
F_{32}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{33}\! \left(x \right)+F_{39}\! \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_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{52}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \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_{19}\! \left(x \right)+F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{39}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{36}\! \left(x \right)
F_{51}\! \left(x \right) = F_{48}\! \left(x \right)
F_{52}\! \left(x \right) = F_{47}\! \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_{58}\! \left(x \right)
F_{55}\! \left(x \right) = F_{2}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{22}\! \left(x \right)+F_{43}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{39}\! \left(x \right)+F_{48}\! \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_20(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_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_15(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_53(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_30(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_1(x) + F_4(x))
Eq(F_29(x), F_15(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_23(x))
Eq(F_32(x), 2*F_22(x) + F_33(x) + F_39(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_4(x))
Eq(F_38(x), F_33(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_52(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_19(x) + F_4(x))
Eq(F_46(x), F_42(x) + F_47(x))
Eq(F_47(x), 2*F_22(x) + F_39(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_36(x))
Eq(F_51(x), F_48(x))
Eq(F_52(x), F_47(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_58(x))
Eq(F_55(x), F_2(x) + F_56(x))
Eq(F_56(x), F_22(x) + F_43(x) + F_57(x))
Eq(F_57(x), F_4(x)*F_55(x))
Eq(F_58(x), F_21(x) + F_59(x))
Eq(F_59(x), 2*F_22(x) + F_39(x) + F_48(x))
Pack JSON:
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Specification JSON:
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