0213_0321_1230_2103_3021
Counting sequence:
1, 1, 2, 6, 19, 48, 98, 185, 337, 602, 1072, 1915, 3439, 6212, 11278, 20557, 37581, 68846, 126300, 231919, 426123, 783256, 1440058, 2648033, 4869769, 8956098, 16471944, 30295651, 55721319, 102486316, 188500454, 346705013, 637688453, 1172890326, 2157279924, 3967854551, 7298020355, 13423150080, 24689019922, 45410184969, 83522349249, 153621548074, 282554075872, 519697966411, 955873583199, 1758125617940, 3233697159614, 5947696352413, 10939519121213, 20120912624062, 37008128088076, 68068559823295, 125197600524923, 230274288425320, 423540448762090, 779012337700401, 1432827074875385, 2635379861324946, 4847219273887288, 8915426210073651, 16398025345271383, 30160670829217276, 55474122384546710, 102032818559019205, 187667611772766453, 345174552716315046, 634874983048082788, 1167717147537145767, 2147766683301524467, 3950358813886733264, 7265842644725383106, 13363968141913619801, 24580169600525714481, 45209980387164695034, 83154118129604006288, 152944268117294392091, 281308366634063069007, 517406752880961442276, 951659387632318877550, 1750374507147343362285, 3219440647660623654829, 5921474542440285866638, 10891289697248252854972, 20032204887349162346895, 36844969127037701038187, 67768463711635116208952, 124645637726021979562138, 229259070564694796776577, 421673172002351892514153, 775577880293068668818530, 1426510122860115358074088, 2623761175155535919370755, 4825849178308719946226503, 8876120476324371223633612, 16325730829788627089192262, 30027700484421718259012885, 55229551790534716571798373, 101582983104745061919962230, 186840235379701496750731284, 343652770274981275242448759, 632075988759427833913098211
Generating function in Maple syntax:
-(x^9+2*x^7+10*x^6+3*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}+2 x^{7}+10 x^{6}+3 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 - 2*x**7 - 10*x**6 - 3*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+2*x^7+10*x^6+3*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}+2 x^{7}+10 x^{6}+3 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*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(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)+5/2*n^2-31/2*n+17)
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 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-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}\\+\frac{5 n^{2}}{2}-\frac{31 n}{2}+17 & \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) = 48
a(6) = 98
a(7) = 185
a(8) = 337
a(9) = 602
a(n+3) = -5*n^2+a(n)+a(n+1)+a(n+2)+31*n-24, 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) = 48
a \! \left(6\right) = 98
a \! \left(7\right) = 185
a \! \left(8\right) = 337
a \! \left(9\right) = 602
a \! \left(n +3\right) = -5 n^{2}+a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+31 n -24, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20525/
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[18,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[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]
F[17,x] = F[11,x]
F[18,x] = F[19,x]+F[38,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[13,x]+F[22,x]
F[22,x] = F[19,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[33,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[11,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[27,x]*F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[19,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[19,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[27,x]
F[38,x] = F[24,x]+F[39,x]+F[56,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[47,x]
F[41,x] = F[42,x]+F[7,x]
F[42,x] = F[24,x]+F[43,x]+F[44,x]
F[43,x] = F[4,x]*F[41,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[17,x]+F[46,x]
F[46,x] = F[28,x]
F[47,x] = F[48,x]+F[53,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[19,x]
F[52,x] = F[31,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[23,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[65,x]
F[58,x] = F[19,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[63,x]+F[7,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[62,x]
F[65,x] = F[23,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[14,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[69,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_{18}\! \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_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \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_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{38}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{33}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \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_{19}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{19}\! \left(x \right)+F_{35}\! \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_{27}\! \left(x \right)
F_{38}\! \left(x \right) = F_{24}\! \left(x \right)+F_{39}\! \left(x \right)+F_{56}\! \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_{47}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{24}\! \left(x \right)+F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{17}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{28}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right)
F_{52}\! \left(x \right) = F_{31}\! \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_{23}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{65}\! \left(x \right)
F_{58}\! \left(x \right) = F_{19}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{7}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{23}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{14}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \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_18(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_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(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))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_19(x) + F_38(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_13(x) + F_22(x))
Eq(F_22(x), F_19(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_33(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_11(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_27(x)*F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_19(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_19(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_27(x))
Eq(F_38(x), F_24(x) + F_39(x) + F_56(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_47(x))
Eq(F_41(x), F_42(x) + F_7(x))
Eq(F_42(x), F_24(x) + F_43(x) + F_44(x))
Eq(F_43(x), F_4(x)*F_41(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_17(x) + F_46(x))
Eq(F_46(x), F_28(x))
Eq(F_47(x), F_48(x) + F_53(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_19(x))
Eq(F_52(x), F_31(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_23(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_65(x))
Eq(F_58(x), F_19(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_63(x) + F_7(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_62(x))
Eq(F_65(x), F_23(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_14(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_69(x))
Pack JSON:
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Specification JSON:
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