0123_0132_1302_1320_3102
Counting sequence:
1, 1, 2, 6, 19, 51, 120, 264, 557, 1143, 2300, 4561, 8943, 17378, 33522, 64271, 122595, 232827, 440518, 830774, 1562333, 2930823, 5486074, 10249483, 19116451, 35600950, 66212224, 122998889, 228248015, 423160715, 783862920, 1450942716, 2683924705, 4961705975, 9167685636, 16930928709, 31254747779, 57674629378, 106390930866, 196196358027, 361707681375, 666680029291, 1228517224902, 2263388730866, 4169270850785, 7678778961935, 14140426973346, 26036202274295, 47933958439383, 88239834620046, 162422600023440, 298946023447565, 550183279414395, 1012496719584419, 1863178611748744, 3428409333040016, 6308274243539173, 11606742073648727, 21354720966148316, 39288277919873465, 72280168198857927, 132973091682630306, 244623437647055522, 450011026702770023, 827827837791913707, 1522823470545743099, 2801254395413746134, 5152876100208903534, 9478544200889247829, 17435280273260465879, 32070969109288288698, 58991785364146703107, 108509485454824538843, 199590990478618482502, 367122961010089636592, 675273693721924571185, 1242069906671069444183, 2284601191126990297019, 4202165099158950617912, 7729196662259584194308, 14216552669881389041273, 26148879086272386296791, 48096206207450034375236, 88464218296675320143213, 162713523018296647699451, 299280846478245514287362, 550469866677583717450626, 1012482673709812537949923, 1862263523481835391010503, 3425265317912749564858219, 6300092005851887399690166, 11587752372016896184591594, 21313324591366641254860153, 39201520049637567149424927, 72103170529122159751226114, 132618951966789980750020207, 243925172598588595275752687, 448649793884637870986689182, 825197998973901324705556576, 1517779628392293293719205909, 2791638300011735214562499491
Generating function in Maple syntax:
(x^12+3*x^11+2*x^10-4*x^9-10*x^8-6*x^7+7*x^6+8*x^5+2*x^4-4*x^3-4*x^2+4*x-1)/(x^3+x^2+x-1)/(x-1)^2/(x^2+x-1)^2
Generating function in latex syntax:
\frac{x^{12}+3 x^{11}+2 x^{10}-4 x^{9}-10 x^{8}-6 x^{7}+7 x^{6}+8 x^{5}+2 x^{4}-4 x^{3}-4 x^{2}+4 x -1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2}}
Generating function in sympy syntax:
(x**12 + 3*x**11 + 2*x**10 - 4*x**9 - 10*x**8 - 6*x**7 + 7*x**6 + 8*x**5 + 2*x**4 - 4*x**3 - 4*x**2 + 4*x - 1)/((x - 1)**2*(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-1)^2*(x^2+x-1)^2*F(x)+x^12+3*x^11+2*x^10-4*x^9-10*x^8-6*x^7+7*x^6+8*x^5+2*x^4-4*x^3-4*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+x^{12}+3 x^{11}+2 x^{10}-4 x^{9}-10 x^{8}-6 x^{7}+7 x^{6}+8 x^{5}+2 x^{4}-4 x^{3}-4 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,n = 3,6,1/13200*(((-3600*11^(1/2)+14300*I)*3^(1/2)-10800*I*11^(1/2)+14300)*(17+3*11^(1/2)*3^(1/2))^(1/3)+41800+((4575*11^(1/2)+28325*I)*3^(1/2)-13725*I*11^(1/2)-28325)*(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*(((4575*11^(1/2)-28325*I)*3^(1/2)+13725*I*11^(1/2)-28325)*(17+3*11^(1/2)*3^(1/2))^(2/3)+41800+((-3600*11^(1/2)-14300*I)*3^(1/2)+10800*I*11^(1/2)+14300)*(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*((-9150*11^(1/2)*3^(1/2)+56650)*(17+3*11^(1/2)*3^(1/2))^(2/3)+7200*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-28600*(17+3*11^(1/2)*3^(1/2))^(1/3)+41800)*(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*((1320*n+32208)*5^(1/2)-3960*n-66000)*(-1/2*5^(1/2)-1/2)^(-n)+1/13200*((-1320*n-32208)*5^(1/2)-3960*n-66000)*(1/2*5^(1/2)-1/2)^(-n)-n-1/2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(\left(\left(-3600 \sqrt{11}+14300 \,\mathrm{I}\right) \sqrt{3}-10800 \,\mathrm{I} \sqrt{11}+14300\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+41800+\left(\left(4575 \sqrt{11}+28325 \,\mathrm{I}\right) \sqrt{3}-13725 \,\mathrm{I} \sqrt{11}-28325\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(4575 \sqrt{11}-28325 \,\mathrm{I}\right) \sqrt{3}+13725 \,\mathrm{I} \sqrt{11}-28325\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+41800+\left(\left(-3600 \sqrt{11}-14300 \,\mathrm{I}\right) \sqrt{3}+10800 \,\mathrm{I} \sqrt{11}+14300\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(-9150 \sqrt{11}\, \sqrt{3}+56650\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+7200 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-28600 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+41800\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(1320 n +32208\right) \sqrt{5}-3960 n -66000\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{13200}+\\\frac{\left(\left(-1320 n -32208\right) \sqrt{5}-3960 n -66000\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{13200}-n -\frac{1}{2} & \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) = 1143
a(10) = 2300
a(11) = 4561
a(12) = 8943
a(n+4) = 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)+1/2*n-3/4, n >= 13
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) = 1143
a \! \left(10\right) = 2300
a \! \left(11\right) = 4561
a \! \left(12\right) = 8943
a \! \left(n +4\right) = \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}+\frac{n}{2}-\frac{3}{4}, \quad n \geq 13
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16572/
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[24,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[18,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[4,x]
F[18,x] = F[17,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[4,x]
F[23,x] = F[18,x]
F[24,x] = F[25,x]+F[36,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[1,x]+F[12,x]
F[29,x] = F[25,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[25,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[25,x]*F[4,x]
F[36,x] = F[19,x]+F[37,x]+F[53,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[43,x]
F[39,x] = F[40,x]+F[7,x]
F[40,x] = F[19,x]+F[20,x]+F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[11,x]
F[43,x] = F[36,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[19,x]+F[35,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[18,x]+F[4,x]
F[51,x] = F[47,x]+F[52,x]
F[52,x] = F[45,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[25,x]+F[47,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_{24}\! \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_{18}\! \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_{10}\! \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_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{18}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{53}\! \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_{43}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{7}\! \left(x \right)
F_{40}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right)
F_{43}\! \left(x \right) = F_{36}\! \left(x \right)+F_{44}\! \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_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{19}\! \left(x \right)+F_{35}\! \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_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{47}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{45}\! \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_{55}\! \left(x \right) = F_{25}\! \left(x \right)+F_{47}\! \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_24(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_18(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_4(x))
Eq(F_18(x), F_17(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_18(x))
Eq(F_24(x), F_25(x) + F_36(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_1(x) + F_12(x))
Eq(F_29(x), F_25(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_25(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_25(x)*F_4(x))
Eq(F_36(x), F_19(x) + F_37(x) + F_53(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_43(x))
Eq(F_39(x), F_40(x) + F_7(x))
Eq(F_40(x), F_19(x) + F_20(x) + F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_11(x))
Eq(F_43(x), F_36(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_19(x) + F_35(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_18(x) + F_4(x))
Eq(F_51(x), F_47(x) + F_52(x))
Eq(F_52(x), F_45(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_25(x) + F_47(x))
Pack JSON:
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Specification JSON:
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