0231_0312_0321_1230_2103
Counting sequence:
1, 1, 2, 6, 19, 52, 128, 305, 716, 1658, 3801, 8662, 19662, 44501, 100508, 226672, 510677, 1149664, 2586800, 5818195, 13082604, 29411292, 66110755, 148588644, 333938962, 750456473, 1686426130, 3789633292, 8515662631, 19135222338, 42997590314, 96616546543, 217098383572, 487820452094, 1096130392443, 2462995233350, 5534320438220, 12435538128463, 27942453904078, 62786210351792, 141079473774877, 317002926146542, 712299475160916, 1600522984287313, 3596343458627566, 8080911949013914, 18157646836039027, 40799866148767728, 91676473634534844, 205995177018075993, 462866978408170976, 1040052687522412574, 2336977220643701481, 5251140221932955386, 11799205092727291850, 26512573374015240597, 59573211921836312824, 133859792615243472556, 300780224571602410661, 675847031120189272532, 1518614496273609167972, 3412295802456254174099, 7667359075503248904720, 17228399465986912452928, 38711862219255909934923, 86984764852028343353864, 195452992494593634423974, 439178887682260512436945, 986826003104655076348310, 2217377901554846601283960, 4982402918787526097457959, 11195357736438551621404606, 25155740490778528971514926, 56524435800290137601061255, 127009254356671052918849296, 285387204025686568967946138, 641259226612337708120998003, 1440896402901107441659111754, 3237664828400866186706611960, 7274966833109918780554693159, 16346702091751013143995578602, 36730706188261912393281285972, 82533147635247169499284197029, 185450299367138322108576888634, 416703040181477235533160215176, 936323232095194041746339436017, 2103899205005290749197992039482, 4727418601925210474960619454766, 10622413176761993580005077508867, 23868345750446294978756424583852, 53631686075733237078173135368504, 120509304725157027663697088109521, 270781949775611113551026185447924, 608441518200662385920391092378626, 1367155681451805342085141349281641, 3071970931328704809129492441421726, 6902656025908621911370610042860198, 15510127301694255479322290916901957, 34850939697968609587369294077397588, 78309350671723146458710583410033608, 175959513739721122267972079003914725
Generating function in Maple syntax:
(x^8-3*x^7-6*x^6+2*x^5+2*x^4+2*x^3+2*x^2-3*x+1)/(x-1)/(x^2+1)/(x^2+x-1)/(x^3-x^2-2*x+1)
Generating function in latex syntax:
\frac{x^{8}-3 x^{7}-6 x^{6}+2 x^{5}+2 x^{4}+2 x^{3}+2 x^{2}-3 x +1}{\left(x -1\right) \left(x^{2}+1\right) \left(x^{2}+x -1\right) \left(x^{3}-x^{2}-2 x +1\right)}
Generating function in sympy syntax:
(x**8 - 3*x**7 - 6*x**6 + 2*x**5 + 2*x**4 + 2*x**3 + 2*x**2 - 3*x + 1)/((x - 1)*(x**2 + 1)*(x**2 + x - 1)*(x**3 - x**2 - 2*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2+1)*(x^2+x-1)*(x^3-x^2-2*x+1)*F(x)-x^8+3*x^7+6*x^6-2*x^5-2*x^4-2*x^3-2*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}+1\right) \left(x^{2}+x -1\right) \left(x^{3}-x^{2}-2 x +1\right) F \! \left(x \right)-x^{8}+3 x^{7}+6 x^{6}-2 x^{5}-2 x^{4}-2 x^{3}-2 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/38220*((125*I*3^(1/2)+765)*(-28+84*I*3^(1/2))^(2/3)-2170*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)-1890*(-28+84*I*3^(1/2))^(1/3)+41160)*(1/168*(I*3^(1/2)+5)*(-28+84*I*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/38220*((320*I*3^(1/2)-570)*(-28+84*I*3^(1/2))^(2/3)+2030*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)-2310*(-28+84*I*3^(1/2))^(1/3)+41160)*(1/84*(I*3^(1/2)-2)*(-28+84*I*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/38220*((-445*I*3^(1/2)-195)*(-28+84*I*3^(1/2))^(2/3)+140*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)+4200*(-28+84*I*3^(1/2))^(1/3)+41160)*(1/6*(-28+84*I*3^(1/2))^(1/3)+1/3-1/168*(-28+84*I*3^(1/2))^(2/3)-1/56*I*(-28+84*I*3^(1/2))^(2/3)*3^(1/2))^(-n)+1/38220*(11466*5^(1/2)-34398)*(-1/2*5^(1/2)-1/2)^(-n)+1/38220*(-11466*5^(1/2)-34398)*(1/2*5^(1/2)-1/2)^(-n)-28/65*cos(1/2*n*Pi)+36/65*sin(1/2*n*Pi)-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(125 \,\mathrm{I} \sqrt{3}+765\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-2170 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-1890 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+41160\right) \left(\frac{\left(\mathrm{I} \sqrt{3}+5\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{38220}\\+\\\frac{\left(\left(320 \,\mathrm{I} \sqrt{3}-570\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+2030 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-2310 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+41160\right) \left(\frac{\left(\mathrm{I} \sqrt{3}-2\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{84}+\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{38220}\\+\\\frac{\left(\left(-445 \,\mathrm{I} \sqrt{3}-195\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+140 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}+4200 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+41160\right) \left(\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}} \sqrt{3}}{56}\right)^{-n}}{38220}\\+\frac{\left(11466 \sqrt{5}-34398\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{38220}+\\\frac{\left(-11466 \sqrt{5}-34398\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{38220}-\frac{28 \cos \left(\frac{n \pi}{2}\right)}{65}+\frac{36 \sin \left(\frac{n \pi}{2}\right)}{65}-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) = 52
a(6) = 128
a(7) = 305
a(8) = 716
a(n) = 3*a(n+2)+a(n+4)+a(n+5)-3*a(n+6)+a(n+7)+2, 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) = 52
a \! \left(6\right) = 128
a \! \left(7\right) = 305
a \! \left(8\right) = 716
a \! \left(n \right) = 3 a \! \left(n +2\right)+a \! \left(n +4\right)+a \! \left(n +5\right)-3 a \! \left(n +6\right)+a \! \left(n +7\right)+2, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20617/
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[13,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[6,x]
F[10,x] = F[11,x]+F[4,x]
F[11,x] = F[12,x]
F[12,x] = F[4,x]*F[7,x]
F[13,x] = F[14,x]+F[2,x]
F[14,x] = F[15,x]+F[16,x]+F[46,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[24,x]
F[18,x] = F[19,x]+F[7,x]
F[19,x] = F[15,x]+F[20,x]+F[22,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[18,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[10,x]
F[24,x] = F[25,x]+F[32,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[2,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[2,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[15,x]+F[31,x]+F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[39,x]+F[4,x]
F[39,x] = F[15,x]+F[40,x]+F[42,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[38,x]
F[42,x] = x^2
F[43,x] = F[30,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[35,x]*F[4,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[56,x]
F[48,x] = F[2,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]+F[7,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[52,x]
F[56,x] = F[35,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[11,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[60,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_{13}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{2}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{46}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \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_{2}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \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_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \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_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = x^{2}
F_{43}\! \left(x \right) = F_{30}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{56}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{7}\! \left(x \right)
F_{53}\! \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_{52}\! \left(x \right)
F_{56}\! \left(x \right) = F_{35}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{11}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{60}\! \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_13(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_6(x))
Eq(F_10(x), F_11(x) + F_4(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_4(x)*F_7(x))
Eq(F_13(x), F_14(x) + F_2(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_46(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_24(x))
Eq(F_18(x), F_19(x) + F_7(x))
Eq(F_19(x), F_15(x) + F_20(x) + F_22(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_18(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_10(x))
Eq(F_24(x), F_25(x) + F_32(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_2(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_2(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_15(x) + F_31(x) + F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_43(x))
Eq(F_38(x), F_39(x) + F_4(x))
Eq(F_39(x), F_15(x) + F_40(x) + F_42(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_38(x))
Eq(F_42(x), x**2)
Eq(F_43(x), F_30(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_35(x)*F_4(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_56(x))
Eq(F_48(x), F_2(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x) + F_7(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_52(x))
Eq(F_56(x), F_35(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_11(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_60(x))
Pack JSON:
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Specification JSON:
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