0321_1302_2301
Counting sequence:
1, 1, 2, 6, 21, 75, 267, 948, 3363, 11928, 42306, 150051, 532203, 1887627, 6695070, 23746197, 84223446, 298725258, 1059524205, 3757939815, 13328729619, 47274581820, 167674350831, 594710452092, 2109329900934, 7481409844611, 26535201173727, 94115536503771, 333810705010326, 1183965909550185, 4199311927200282, 14894196293731278, 52827007634105253, 187367796189739443, 664559524025259531, 2357071865890176756, 8360105573868544923, 29651775246077442672, 105169458384846474090, 373016957169373349187, 1323023361276020229411, 4692523438518400474203, 16643527896444001042110, 59031568934937663053565, 209374247611579510437774, 742612408137603576993810, 2633911261823353811503101, 9342004603125886126865967, 33134392669102118496292659, 117521669512225337154776796, 416828005355891633176725927, 1478413187713412498443846836, 5243662914968388704623209006, 18598319464629143431983141795, 65964859396475622427507995447, 233965368939508417659510753483, 829832646712575784028455220118, 2943265598115255281467735566033, 10439228217119346956146478769906, 37026045437042930444011346113686, 131324654676844336327581064647093, 465784685413322139842276682863067, 1652053635316673472577931064773835, 5859534028133967532818738420632052, 20782702385009763248152935763106451, 73712468662195031461586041486061448, 261444731085318480417468808113564690, 927296950608451503062286598940774115, 3288953772516931727191378270480666779, 11665321352189905058835578183099383275, 41374774977673289440216481089435238046, 146748808092777812029643071160021890917, 520491354654419038683423203812796176134, 1846088249648448189559907035372155249306, 6547739544594060151787925167464064862477, 23223642289042868737630453236525781344951, 82370038926595327346104436283960983297811, 292151559532501604760103237503042423006780, 1036208490970064665936780929521023559281887, 3675243214435099683161598531438746208262188, 13035419804952615234467616792576716954639958, 46234265211062672071178835562814178201472323, 163984536868902967998573272652926827904308431, 581623352492996653269196200265479221176628091, 2062912336884753474568077872098277673061706838, 7316775180072496357220931976857293521961690937, 25951271936532950898503781252493239234968097930, 92044445612884036772897548017617268062255464542, 326464921985439967545127848503698785467353807941, 1157911752059494469783765298170270036668140546243, 4106902565223484653966752043882009274723346277195, 14566437079716776353766551975899522510009949532340, 51664505263421457254383543510716367011714091637259, 183244611534477082111218661829830727822680799823840, 649935337331006996091091704012554004573149297892666, 2305202533238436859551297179615934756964803965432067, 8176134476809573567543591659493841940853166439275859, 28999263196610267668683450331811263176196456297464891, 102855116721909207516809593240941068506757694747965502, 364808407860313152274578039192319289299169235272865741, 1293909128559941319278082987511383824983889376768930366
Generating function in Maple syntax:
(x^3-3*x^2+4*x-1)/(3*x^3-6*x^2+5*x-1)
Generating function in latex syntax:
\frac{x^{3}-3 x^{2}+4 x -1}{3 x^{3}-6 x^{2}+5 x -1}
Generating function in sympy syntax:
(x**3 - 3*x**2 + 4*x - 1)/(3*x**3 - 6*x**2 + 5*x - 1)
Implicit equation for the generating function in Maple syntax:
(3*x^3-6*x^2+5*x-1)*F(x)-x^3+3*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(3 x^{3}-6 x^{2}+5 x -1\right) F \! \left(x \right)-x^{3}+3 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/2088*(-(I*3^(1/2)-1)*(29^(1/2)+29)*2^(2/3)*(5+29^(1/2))^(1/3)+464-17*2^(1/3)*(29^(1/2)-87/17)*(1+I*3^(1/2))*(5+29^(1/2))^(2/3))*(1/24*(-2*I*3^(1/2)+2)*(20+4*29^(1/2))^(1/3)+2/3-1/24*(29^(1/2)-5)*2^(1/3)*(1+I*3^(1/2))*(5+29^(1/2))^(2/3))^(-n)+1/2088*((29^(1/2)+29)*(1+I*3^(1/2))*2^(2/3)*(5+29^(1/2))^(1/3)+464+17*2^(1/3)*(29^(1/2)-87/17)*(I*3^(1/2)-1)*(5+29^(1/2))^(2/3))*(1/24*(2*I*3^(1/2)+2)*(20+4*29^(1/2))^(1/3)+2/3+1/24*(I*3^(1/2)-1)*(29^(1/2)-5)*2^(1/3)*(5+29^(1/2))^(2/3))^(-n)+17/1044*(-1/17*2^(2/3)*(29^(1/2)+29)*(5+29^(1/2))^(1/3)+232/17+2^(1/3)*(29^(1/2)-87/17)*(5+29^(1/2))^(2/3))*(-1/6*(20+4*29^(1/2))^(1/3)+2/3+1/12*2^(1/3)*(29^(1/2)-5)*(5+29^(1/2))^(2/3))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{29}+29\right) 2^{\frac{2}{3}} \left(5+\sqrt{29}\right)^{\frac{1}{3}}+464-17 \,2^{\frac{1}{3}} \left(\sqrt{29}-\frac{87}{17}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(5+\sqrt{29}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}+2\right) \left(20+4 \sqrt{29}\right)^{\frac{1}{3}}}{24}+\frac{2}{3}-\frac{\left(\sqrt{29}-5\right) 2^{\frac{1}{3}} \left(1+\mathrm{I} \sqrt{3}\right) \left(5+\sqrt{29}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{2088}\\+\\\frac{\left(\left(\sqrt{29}+29\right) \left(1+\mathrm{I} \sqrt{3}\right) 2^{\frac{2}{3}} \left(5+\sqrt{29}\right)^{\frac{1}{3}}+464+17 \,2^{\frac{1}{3}} \left(\sqrt{29}-\frac{87}{17}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(5+\sqrt{29}\right)^{\frac{2}{3}}\right) \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}+2\right) \left(20+4 \sqrt{29}\right)^{\frac{1}{3}}}{24}+\frac{2}{3}+\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{29}-5\right) 2^{\frac{1}{3}} \left(5+\sqrt{29}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{2088}\\+\\\frac{17 \left(-\frac{2^{\frac{2}{3}} \left(\sqrt{29}+29\right) \left(5+\sqrt{29}\right)^{\frac{1}{3}}}{17}+\frac{232}{17}+2^{\frac{1}{3}} \left(\sqrt{29}-\frac{87}{17}\right) \left(5+\sqrt{29}\right)^{\frac{2}{3}}\right) \left(-\frac{\left(20+4 \sqrt{29}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}+\frac{2^{\frac{1}{3}} \left(\sqrt{29}-5\right) \left(5+\sqrt{29}\right)^{\frac{2}{3}}}{12}\right)^{-n}}{1044} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+3) = 3*a(n)-6*a(n+1)+5*a(n+2), n >= 4
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(n +3\right) = 3 a \! \left(n \right)-6 a \! \left(n +1\right)+5 a \! \left(n +2\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/383/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[0,x]*F[12,x]*F[7,x]
F[7,x] = F[32,x]+F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[8,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[22,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[19,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[12,x]+F[25,x]
F[25,x] = F[18,x]+F[22,x]+F[26,x]
F[26,x] = F[10,x]*F[12,x]
F[27,x] = F[28,x]+F[30,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[14,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[21,x]
F[32,x] = F[33,x]+F[41,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[1,x]+F[34,x]
F[37,x] = F[18,x]+F[38,x]+F[40,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[10,x]+F[37,x]
F[40,x] = F[12,x]*F[32,x]
F[41,x] = F[42,x]+F[47,x]
F[42,x] = F[18,x]+F[43,x]+F[45,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[14,x]+F[42,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[34,x]+F[42,x]
F[47,x] = F[18,x]+F[48,x]+F[51,x]+F[53,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[21,x]+F[50,x]
F[50,x] = 2*F[18,x]+F[48,x]+F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[37,x]+F[50,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[60,x]
F[55,x] = F[56,x]+F[58,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[34,x]
F[58,x] = 2*F[18,x]+F[53,x]+F[59,x]
F[59,x] = F[12,x]*F[37,x]
F[60,x] = F[61,x]+F[63,x]
F[61,x] = F[62,x]
F[62,x] = F[12,x]*F[42,x]
F[63,x] = F[64,x]
F[64,x] = F[12,x]*F[50,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_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{32}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{41}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{14}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{34}\! \left(x \right)+F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{21}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{37}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{60}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_0(x)*F_12(x)*F_7(x))
Eq(F_7(x), F_32(x) + F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_8(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_22(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_19(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_12(x) + F_25(x))
Eq(F_25(x), F_18(x) + F_22(x) + F_26(x))
Eq(F_26(x), F_10(x)*F_12(x))
Eq(F_27(x), F_28(x) + F_30(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_14(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_21(x))
Eq(F_32(x), F_33(x) + F_41(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_1(x) + F_34(x))
Eq(F_37(x), F_18(x) + F_38(x) + F_40(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_10(x) + F_37(x))
Eq(F_40(x), F_12(x)*F_32(x))
Eq(F_41(x), F_42(x) + F_47(x))
Eq(F_42(x), F_18(x) + F_43(x) + F_45(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_14(x) + F_42(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_34(x) + F_42(x))
Eq(F_47(x), F_18(x) + F_48(x) + F_51(x) + F_53(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_21(x) + F_50(x))
Eq(F_50(x), 2*F_18(x) + F_48(x) + F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_37(x) + F_50(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_60(x))
Eq(F_55(x), F_56(x) + F_58(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_34(x))
Eq(F_58(x), 2*F_18(x) + F_53(x) + F_59(x))
Eq(F_59(x), F_12(x)*F_37(x))
Eq(F_60(x), F_61(x) + F_63(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_12(x)*F_42(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_12(x)*F_50(x))
Pack JSON:
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Specification JSON:
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