0213_0321_1230_1302
Counting sequence:
1, 1, 2, 6, 20, 62, 183, 526, 1486, 4144, 11439, 31319, 85182, 230420, 620475, 1664474, 4450753, 11868662, 31575721, 83836017, 222204779, 588061123, 1554259617, 4103267652, 10821913860, 28516766670, 75087073258, 197578299465, 519586282746, 1365686915986, 3587950497994, 9422518399525, 24736220466817, 64917758689358, 170323024955610, 446762608629903, 1171618355271118, 3071936504643999, 8053127484820916, 21108220387714783, 55319828387152009, 144963536350594941, 379831780415432787, 995138555703224521, 2606994411192057830, 6829123749502157902, 17887999754013929668, 46852596643877571666, 122710986809862788371, 321376134336466435234, 841640055718653377174, 2204061606907576995284, 5771747978780849427767, 15113979462108237007135, 39576692952098198411810, 103631215976279609063664, 271352096766584973368063, 710506769073376179954602, 1860358127705650734895581, 4871009118089940136692382, 12753695598948948543176689, 33392463705070621041302413, 87429242354480867602094727, 228908158192783188600139967, 599325209076980128209536145, 1569137156766887025413149160, 4108248265538058782996689668, 10755984254186074599719104430, 28160580019136495004038301406, 73727791145209207658711242301, 193027525012554645011428582054, 505365783518790013988820766290, 1323095396572680019962421633502, 3463979851629236858733448515973, 9068982352173597035341407488745, 23743288466636144681938603826418, 62161629890681462169045930852078, 162743337404617502910721593136539, 426072418496650169937190837930726, 1115483301050298480010593606823323, 2920399297401392703761196466702644, 7645765299638870109427401862812631, 20017014484440859664760757048160197, 52405552198237801482931907406050161, 137200279186568528975940989759815015, 359196766384173194864694807320529617, 940393462926029408091819657487910778, 2461991626305035252184091374697168806, 6445600022824985721252887390140354756, 16874851697815487936878891372512377574, 44179055627797477642485209820242672159, 115662548952649720972064799136713116374, 302809134672663580137274845126485803102, 792766118415907151620582009848362610600, 2075492157508805257521294751210776206895, 5433717181653130399835694054485841035479, 14225675259561522645250137984373641936998, 37243345495212751470719811560404194015292, 97504447003941421943688262348594352778691, 255270194925953893724336039689233778888122, 668306601344399330902540209688930011487849
Generating function in Maple syntax:
(2*x^5-3*x^4+4*x^3-8*x^2+5*x-1)/(x^2-3*x+1)/(x^3-2*x^2+3*x-1)
Generating function in latex syntax:
\frac{2 x^{5}-3 x^{4}+4 x^{3}-8 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(2*x**5 - 3*x**4 + 4*x**3 - 8*x**2 + 5*x - 1)/((x**2 - 3*x + 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^3-2*x^2+3*x-1)*F(x)-2*x^5+3*x^4-4*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)-2 x^{5}+3 x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1/20*(11+3*23^(1/2)*3^(1/2))^(1/3)*2^(1/3)*(((-7/10*I+59/690*23^(1/2))*3^(1/2)+59/230*I*23^(1/2)-7/10)*(11+3*23^(1/2)*3^(1/2))^(1/3)+2^(1/3)*((I-8/69*23^(1/2))*3^(1/2)+8/23*I*23^(1/2)-1))*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/20*(11+3*23^(1/2)*3^(1/2))^(1/3)*2^(1/3)*(((-7/10*I-59/690*23^(1/2))*3^(1/2)+59/230*I*23^(1/2)+7/10)*(11+3*23^(1/2)*3^(1/2))^(1/3)+2^(1/3)*((I+8/69*23^(1/2))*3^(1/2)+8/23*I*23^(1/2)+1))*(-11/600*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*((-160*3^(1/2)*2^(2/3)*23^(1/2)-1380*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+118*(11+3*23^(1/2)*3^(1/2))^(2/3)*2^(1/3)*(23^(1/2)*3^(1/2)-483/59))*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*(9660*5^(1/2)-6900)*(3/2-1/2*5^(1/2))^(-n)+1/13800*(-9660*5^(1/2)-6900)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -\frac{\left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}} 2^{\frac{1}{3}} \left(\left(\left(-\frac{7 \,\mathrm{I}}{10}+\frac{59 \sqrt{23}}{690}\right) \sqrt{3}+\frac{59 \,\mathrm{I} \sqrt{23}}{230}-\frac{7}{10}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{8 \sqrt{23}}{69}\right) \sqrt{3}+\frac{8 \,\mathrm{I} \sqrt{23}}{23}-1\right)\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{20}\\+\\\frac{\left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}} 2^{\frac{1}{3}} \left(\left(\left(-\frac{7 \,\mathrm{I}}{10}-\frac{59 \sqrt{23}}{690}\right) \sqrt{3}+\frac{59 \,\mathrm{I} \sqrt{23}}{230}+\frac{7}{10}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{8 \sqrt{23}}{69}\right) \sqrt{3}+\frac{8 \,\mathrm{I} \sqrt{23}}{23}+1\right)\right) \left(-\frac{11 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{20}\\+\\\frac{\left(\left(-160 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{23}-1380 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+118 \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}} 2^{\frac{1}{3}} \left(\sqrt{23}\, \sqrt{3}-\frac{483}{59}\right)\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{13800}\\+\frac{\left(9660 \sqrt{5}-6900\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{13800}+\frac{\left(-9660 \sqrt{5}-6900\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{13800} & \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) = 62
a(n+5) = a(n)-5*a(n+1)+10*a(n+2)-12*a(n+3)+6*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) = 62
a \! \left(n +5\right) = a \! \left(n \right)-5 a \! \left(n +1\right)+10 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15734/
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[12,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[39,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[13,x]
F[24,x] = F[18,x]+F[25,x]
F[25,x] = F[26,x]+F[27,x]+F[31,x]
F[26,x] = 0
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[22,x]
F[30,x] = F[25,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[18,x]+F[33,x]
F[33,x] = F[26,x]+F[34,x]+F[38,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[12,x]
F[37,x] = F[33,x]
F[38,x] = F[12,x]*F[18,x]
F[39,x] = F[26,x]+F[40,x]+F[54,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[46,x]
F[42,x] = F[43,x]+F[6,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[6,x]
F[46,x] = F[39,x]+F[47,x]
F[47,x] = 2*F[26,x]+F[48,x]+F[52,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[43,x]
F[51,x] = F[47,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[39,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[17,x]+F[56,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[26,x]+F[34,x]+F[58,x]
F[58,x] = F[12,x]*F[32,x]
F[59,x] = F[60,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[56,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_{12}\! \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_{39}\! \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_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \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_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{22}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{26}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
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_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{39}\! \left(x \right) = F_{26}\! \left(x \right)+F_{40}\! \left(x \right)+F_{54}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{6}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{6}\! \left(x \right)
F_{46}\! \left(x \right) = F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{48}\! \left(x \right)+F_{52}\! \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_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{43}\! \left(x \right)
F_{51}\! \left(x \right) = F_{47}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{39}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{17}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{26}\! \left(x \right)+F_{34}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{56}\! \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_12(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_39(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_24(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_13(x))
Eq(F_24(x), F_18(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_31(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_22(x))
Eq(F_30(x), F_25(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_18(x) + F_33(x))
Eq(F_33(x), F_26(x) + F_34(x) + F_38(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_12(x))
Eq(F_37(x), F_33(x))
Eq(F_38(x), F_12(x)*F_18(x))
Eq(F_39(x), F_26(x) + F_40(x) + F_54(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_46(x))
Eq(F_42(x), F_43(x) + F_6(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_6(x))
Eq(F_46(x), F_39(x) + F_47(x))
Eq(F_47(x), 2*F_26(x) + F_48(x) + F_52(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_43(x))
Eq(F_51(x), F_47(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_39(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_17(x) + F_56(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_26(x) + F_34(x) + F_58(x))
Eq(F_58(x), F_12(x)*F_32(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_56(x))
Pack JSON:
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Specification JSON:
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