0132_0321_1230_2103_3021
Counting sequence:
1, 1, 2, 6, 19, 47, 93, 173, 315, 567, 1021, 1845, 3347, 6095, 11133, 20381, 37371, 68599, 126013, 231589, 425747, 782831, 1439581, 2647501, 4869179, 8955447, 16471229, 30294869, 55720467, 102485391, 188499453, 346703933, 637687291, 1172889079, 2157278589, 3967853125, 7298018835, 13423148463, 24689018205, 45410183149, 83522347323, 153621546039, 282554073725, 519697964149, 955873580819, 1758125615439, 3233697156989, 5947696349661, 10939519118331, 20120912621047, 37008128084925, 68068559820005, 125197600521491, 230274288421743, 423540448758365, 779012337696525, 1432827074871355, 2635379861320759, 4847219273882941, 8915426210069141, 16398025345266707, 30160670829212431, 55474122384541693, 102032818559014013, 187667611772761083, 345174552716309495, 634874983048077053, 1167717147537139845, 2147766683301518355, 3950358813886726959, 7265842644725376605, 13363968141913613101, 24580169600525707579, 45209980387164687927, 83154118129603998973, 152944268117294384565, 281308366634063061267, 517406752880961434319, 951659387632318869373, 1750374507147343353885, 3219440647660623646203, 5921474542440285857783, 10891289697248252845885, 20032204887349162337573, 36844969127037701028627, 67768463711635116199151, 124645637726021979552093, 229259070564694796766285, 421673172002351892503611, 775577880293068668807735, 1426510122860115358063037, 2623761175155535919359445, 4825849178308719946214931, 8876120476324371223621775, 16325730829788627089180157, 30027700484421718259000509, 55229551790534716571785723, 101582983104745061919949303, 186840235379701496750718077, 343652770274981275242435269, 632075988759427833913084435
Generating function in Maple syntax:
(x^9+3*x^8+x^7-11*x^6-4*x^5+4*x^4+x^3+3*x^2-3*x+1)/(x^3+x^2+x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{9}+3 x^{8}+x^{7}-11 x^{6}-4 x^{5}+4 x^{4}+x^{3}+3 x^{2}-3 x +1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**9 + 3*x**8 + x**7 - 11*x**6 - 4*x**5 + 4*x**4 + x**3 + 3*x**2 - 3*x + 1)/((x - 1)**3*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^3+x^2+x-1)*(x-1)^3*F(x)+x^9+3*x^8+x^7-11*x^6-4*x^5+4*x^4+x^3+3*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}+3 x^{8}+x^{7}-11 x^{6}-4 x^{5}+4 x^{4}+x^{3}+3 x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,n = 3,6,1/132*(((55*I-9*11^(1/2))*3^(1/2)-27*I*11^(1/2)+55)*(17+3*11^(1/2)*3^(1/2))^(1/3)+176+((-11*I-3*11^(1/2))*3^(1/2)+9*I*11^(1/2)+11)*(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)+1/132*(((11*I-3*11^(1/2))*3^(1/2)-9*I*11^(1/2)+11)*(17+3*11^(1/2)*3^(1/2))^(2/3)+176+((-9*11^(1/2)-55*I)*3^(1/2)+27*I*11^(1/2)+55)*(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)+1/132*((6*11^(1/2)*3^(1/2)-22)*(17+3*11^(1/2)*3^(1/2))^(2/3)+18*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)-110*(17+3*11^(1/2)*3^(1/2))^(1/3)+176)*(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)+n^2-3*n-9)
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(55 \,\mathrm{I}-9 \sqrt{11}\right) \sqrt{3}-27 \,\mathrm{I} \sqrt{11}+55\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+176+\left(\left(-11 \,\mathrm{I}-3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}+11\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}}{132}\\+\\\frac{\left(\left(\left(11 \,\mathrm{I}-3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}+11\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+176+\left(\left(-9 \sqrt{11}-55 \,\mathrm{I}\right) \sqrt{3}+27 \,\mathrm{I} \sqrt{11}+55\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}}{132}\\+\\\frac{\left(\left(6 \sqrt{11}\, \sqrt{3}-22\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+18 \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}-110 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+176\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}}{132}\\+n^{2}-3 n -9 & \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) = 47
a(6) = 93
a(7) = 173
a(8) = 315
a(9) = 567
a(n+3) = -2*n^2+a(n)+a(n+1)+a(n+2)+6*n+22, n >= 10
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) = 47
a \! \left(6\right) = 93
a \! \left(7\right) = 173
a \! \left(8\right) = 315
a \! \left(9\right) = 567
a \! \left(n +3\right) = -2 n^{2}+a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+6 n +22, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19502/
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[21,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[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]+F[18,x]
F[21,x] = F[22,x]+F[37,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[15,x]+F[27,x]+F[32,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[17,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[22,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[22,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[17,x]
F[37,x] = F[15,x]+F[38,x]+F[57,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[49,x]
F[40,x] = F[41,x]+F[7,x]
F[41,x] = F[15,x]+F[16,x]+F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[20,x]+F[44,x]
F[44,x] = F[18,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[18,x]+F[48,x]
F[48,x] = F[46,x]
F[49,x] = F[50,x]+F[54,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[31,x]+F[53,x]
F[53,x] = F[30,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[26,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[33,x]+F[59,x]
F[59,x] = F[26,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[14,x]+F[45,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_{21}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \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_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{37}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{22}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{17}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{57}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{7}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{20}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{18}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{54}\! \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_{31}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{30}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{26}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{33}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{26}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{14}\! \left(x \right)+F_{45}\! \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_21(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_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x) + F_18(x))
Eq(F_21(x), F_22(x) + F_37(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_15(x) + F_27(x) + F_32(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_17(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_22(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_22(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_17(x))
Eq(F_37(x), F_15(x) + F_38(x) + F_57(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_49(x))
Eq(F_40(x), F_41(x) + F_7(x))
Eq(F_41(x), F_15(x) + F_16(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_20(x) + F_44(x))
Eq(F_44(x), F_18(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_18(x) + F_48(x))
Eq(F_48(x), F_46(x))
Eq(F_49(x), F_50(x) + F_54(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_31(x) + F_53(x))
Eq(F_53(x), F_30(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_26(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_33(x) + F_59(x))
Eq(F_59(x), F_26(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_14(x) + F_45(x))
Pack JSON:
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Specification JSON:
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