0213_0321_1302_2130
Counting sequence:
1, 1, 2, 6, 20, 64, 195, 575, 1661, 4731, 13331, 37236, 103247, 284512, 779889, 2128130, 5784422, 15668745, 42315569, 113974704, 306257558, 821191588, 2197736829, 5871676346, 15663004736, 41723159604, 111000170750, 294960168213, 782960578013, 2076309554509, 5501167266327, 14563289756681, 38524260114536, 101836833011601, 269026110601490, 710270947726851, 1874187197182898, 4942864872610942, 13029790359589095, 34332368211779021, 90425570289910142, 238074205503304212, 626582803547470886, 1648542688264704501, 4335979995647743298, 11401151092637353018, 29970420235470911016, 78763932660010924988, 206947196545347550355, 543620657807616532695, 1427719155338619477041, 3748916774297144641803, 9842158142097592695167, 25834578855113585001952, 67802168261257785539835, 177918560188762841483264, 466808163764723756901833, 1224613135157709021548622, 3212226703399449796687334, 8424876755922402416630277, 22093986991596474977933741, 57934955408446221842337340, 151902876198560686370443326, 398247055380299976502811588, 1044005447459240918941941185, 2736646996994164274833811346, 7173030742504901922284398484, 18799938949014139113776858976, 49269918113593880305983947906, 129116160432469923293313405133, 338340764456824543961093526357, 886552608853695313353637415825, 2322910994524091807686228655519, 6086110328638469695393748855493, 15945109572718798377608780183660, 41773108742487893140203369065685, 109433120021747015115581726044970, 286671481769894293280350805162635, 750939401300399661766117010769294, 1967029584046460577029281729078358, 5152326109778710047385049565846659, 13495315699510193443444683980298657, 35346853596680793560941507981292402, 92577871027964444893945741497508356, 242467201059958526984153245651247558, 635022066579520561072692121165374121, 1663088006337607699938045679583148978, 4355447635649511546076817123411515166, 11406227598463568524155816082397214476, 29870564558047728964279779346841010840, 78223537229291177212929179509122585635, 204844602846297288119634879444446857407, 536420126602348439518923096794097241733, 1404686633010216289236026334150751634443, 3678307587311849157855366769452148786491, 9631882674090428066119300727226763156796, 25221400109350483695725063874981179086439, 66042327070350379421998503575005558928544, 172930260030798127543295935504488650285873, 452809300679660259060945779558088193353306, 1185647666242328377389467704407887727015638
Generating function in Maple syntax:
(x^5-6*x^4+12*x^3-13*x^2+6*x-1)/(x^2-3*x+1)/(3*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
\frac{x^{5}-6 x^{4}+12 x^{3}-13 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(x**5 - 6*x**4 + 12*x**3 - 13*x**2 + 6*x - 1)/((x**2 - 3*x + 1)*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(3*x^3-5*x^2+4*x-1)*F(x)-x^5+6*x^4-12*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-x^{5}+6 x^{4}-12 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/1350360*(24025*2^(1/3)*((I-45/961*31^(1/2))*3^(1/2)-135/961*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(2/3)-600160+8525*2^(2/3)*((I+27/31*31^(1/2))*3^(1/2)-81/31*I*31^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(1/3))*(47/8712*((I-9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)+1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/1350360*(-8525*((I-27/31*31^(1/2))*3^(1/2)-81/31*I*31^(1/2)+1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)-600160-24025*2^(1/3)*((I+45/961*31^(1/2))*3^(1/2)-135/961*I*31^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(2/3))*(-47/8712*((I+9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)-1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)+1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/1350360*((-14850*3^(1/2)*2^(2/3)*31^(1/2)+17050*2^(2/3))*(47+9*31^(1/2)*3^(1/2))^(1/3)-600160+(2250*31^(1/2)*3^(1/2)*2^(1/3)-48050*2^(1/3))*(47+9*31^(1/2)*3^(1/2))^(2/3))*(1/4356*2^(1/3)*(9*31^(1/2)*3^(1/2)-47)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/18*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/1350360*(540144*5^(1/2)+1350360)*(3/2-1/2*5^(1/2))^(-n)-2/5*(5^(1/2)-5/2)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(24025 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{45 \sqrt{31}}{961}\right) \sqrt{3}-\frac{135 \,\mathrm{I} \sqrt{31}}{961}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}-600160+8525 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{27 \sqrt{31}}{31}\right) \sqrt{3}-\frac{81 \,\mathrm{I} \sqrt{31}}{31}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{47 \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{1350360}\\+\\\frac{\left(-8525 \left(\left(\mathrm{I}-\frac{27 \sqrt{31}}{31}\right) \sqrt{3}-\frac{81 \,\mathrm{I} \sqrt{31}}{31}+1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-600160-24025 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{45 \sqrt{31}}{961}\right) \sqrt{3}-\frac{135 \,\mathrm{I} \sqrt{31}}{961}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{47 \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{1350360}\\+\\\frac{\left(\left(-14850 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{31}+17050 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-600160+\left(2250 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}-48050 \,2^{\frac{1}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{1350360}\\+\frac{\left(540144 \sqrt{5}+1350360\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{1350360}-\frac{2 \left(\sqrt{5}-\frac{5}{2}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{5} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(n+5) = 3*a(n)-14*a(n+1)+22*a(n+2)-18*a(n+3)+7*a(n+4), n >= 6
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) = 20
a \! \left(5\right) = 64
a \! \left(n +5\right) = 3 a \! \left(n \right)-14 a \! \left(n +1\right)+22 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15745/
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[12,x]*F[4,x]
F[4,x] = F[17,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = x
F[13,x] = F[1,x]+F[10,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[9,x]
F[17,x] = F[18,x]+F[32,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[5,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[21,x]+F[25,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[18,x]+F[26,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[25,x]
F[32,x] = F[33,x]+F[34,x]+F[44,x]
F[33,x] = 0
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[40,x]
F[36,x] = F[37,x]+F[6,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[36,x]
F[40,x] = F[32,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[40,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[17,x]+F[46,x]
F[46,x] = F[47,x]+F[58,x]
F[47,x] = F[27,x]+F[33,x]+F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]+F[54,x]
F[50,x] = F[10,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[50,x]
F[54,x] = F[26,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[54,x]
F[58,x] = F[59,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[46,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{17}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{14}\! \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_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{5}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \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_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{44}\! \left(x \right)
F_{33}\! \left(x \right) = 0
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{6}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \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_{47}\! \left(x \right)+F_{58}\! \left(x \right)
F_{47}\! \left(x \right) = F_{27}\! \left(x \right)+F_{33}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{10}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{50}\! \left(x \right)
F_{54}\! \left(x \right) = F_{26}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{54}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{46}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_17(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), x)
Eq(F_13(x), F_1(x) + F_10(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_9(x))
Eq(F_17(x), F_18(x) + F_32(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_5(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_21(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_18(x) + F_26(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_25(x))
Eq(F_32(x), F_33(x) + F_34(x) + F_44(x))
Eq(F_33(x), 0)
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_40(x))
Eq(F_36(x), F_37(x) + F_6(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_36(x))
Eq(F_40(x), F_32(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_40(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_17(x) + F_46(x))
Eq(F_46(x), F_47(x) + F_58(x))
Eq(F_47(x), F_27(x) + F_33(x) + F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_54(x))
Eq(F_50(x), F_10(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_50(x))
Eq(F_54(x), F_26(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_54(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_46(x))
Pack JSON:
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Specification JSON:
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