0132_0321_1023_1032
Counting sequence:
1, 1, 2, 6, 20, 66, 216, 706, 2308, 7546, 24672, 80666, 263740, 862306, 2819336, 9217906, 30138228, 98537866, 322172592, 1053353226, 3443970860, 11260168946, 36815469656, 120369313506, 393551182948, 1286727730586, 4206996000512, 13754903176186, 44972080164380, 147037603129986, 480743978379176, 1571807264455506, 5139072328944468, 16802355479169706, 54935819458756432, 179614356055405546, 587254676803692300, 1920047277966065426, 6277653793563995896, 20524982693965171906, 67107063951100830788, 219408615309635799226, 717363234773171048352, 2345441221065978457306, 7668492410563672282620, 25072372448603095855266, 81974895004853323487816, 268019447494731350950706, 876297849861571625049908, 2865082846971799094323146, 9367476733291944204044272, 30627254091975522783307466, 100136751861969856709743340, 327400198638536746897139506, 1070445047152188463025942936, 3499853096417083324831839906, 11442877641488841404043809828, 37412841371008492567731605466, 122322438752401612708458672192, 399936986179558655938727492026, 1307606311203046770764997162460, 4275259163778314340660723518146, 13978091693863234362225357569256, 45701801906524087968449254047506, 149423450872063277468442390518548, 488544581156407314692872515070186, 1597311575822487602303061396820112, 5222459461564855020070512823106986, 17074992281104771902758612132107980, 55827213891369497755429032852751186, 182528797645293385353581320660894776, 596783533469274591953555408349541506, 1951202168723957207264335737120984068, 6379515669778591823903839364757101306, 20858023239881044857729461064364536032, 68195950287636318373734997637428808346, 222968762770459322569559286086135827900, 729003246698690424872491073560985958626, 2383498599058656368528756431260393967496, 7792922181679472844636671076530889649906, 25479199422939303122705397159464348614388, 83305028344836106693969088270785513442826, 272368359473885463097033503757810787227952, 890516751706912138551637544537815305812106, 2911571985095664378303387843419399888450220, 9519474404209692023752705747579775210820466, 31124215164965599237203934549136531789555416, 101761581417424649634164396640645601302660706, 332712628983221184872551194410332360263617508, 1087814202010541989062119456817665700725598746, 3556642083927353700899153037320956924716863872, 11628550987644231206155356167651495317217894266, 38019906102815947700048204472278441896172182940, 124307255616185804583525055460801195482468776706, 406426406131583837646266320761672446875794346536, 1328823669883409832234586525608843084848115421906, 4344629972371259425166597251071957389730016201428, 14204901692097975879255162068984645198127307233066, 46443364191044944905990029573940394793015411171792, 151848011632628370836527826590951558357139755390826, 496471757341574600386651541779954339445767402514060
Generating function in Maple syntax:
(x^3-x^2+3*x-1)/(2*x^3-3*x^2+4*x-1)
Generating function in latex syntax:
\frac{x^{3}-x^{2}+3 x -1}{2 x^{3}-3 x^{2}+4 x -1}
Generating function in sympy syntax:
(x**3 - x**2 + 3*x - 1)/(2*x**3 - 3*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(2*x^3-3*x^2+4*x-1)*F(x)-x^3+x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)-x^{3}+x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/17100*(((15*I*3^(1/3)-5*3^(5/6))*38^(1/2)-285*I*3^(5/6)+285*3^(1/3))*(9+2*38^(1/2)*3^(1/2))^(1/3)+2850+((-21*I*3^(2/3)-21*3^(1/6))*38^(1/2)+57*I*3^(1/6)+19*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3))*(1/12*(27+6*38^(1/2)*3^(1/2))^(1/3)-1/12*I*3^(5/6)*(9+2*38^(1/2)*3^(1/2))^(1/3)+1/2+1/300*((9*I-2*38^(1/2))*3^(1/6)+(-2*I*38^(1/2)+3)*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3))^(-n)+1/17100*(((-15*I*3^(1/3)-5*3^(5/6))*38^(1/2)+285*I*3^(5/6)+285*3^(1/3))*(9+2*38^(1/2)*3^(1/2))^(1/3)+2850+((21*I*3^(2/3)-21*3^(1/6))*38^(1/2)-57*I*3^(1/6)+19*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3))*(1/12*(27+6*38^(1/2)*3^(1/2))^(1/3)+1/12*I*3^(5/6)*(9+2*38^(1/2)*3^(1/2))^(1/3)+1/2+1/300*((-9*I-2*38^(1/2))*3^(1/6)+(2*I*38^(1/2)+3)*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3))^(-n)-1/30*((-1/57*3^(5/6)*38^(1/2)+3^(1/3))*(9+2*38^(1/2)*3^(1/2))^(1/3)-5+(-7/95*38^(1/2)*3^(1/6)+1/15*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3))*(1/150*(2*38^(1/2)*3^(1/6)-3*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3)-1/6*(27+6*38^(1/2)*3^(1/2))^(1/3)+1/2)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(15 \,\mathrm{I} \,3^{\frac{1}{3}}-5 \,3^{\frac{5}{6}}\right) \sqrt{38}-285 \,\mathrm{I} \,3^{\frac{5}{6}}+285 \,3^{\frac{1}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}+2850+\left(\left(-21 \,\mathrm{I} \,3^{\frac{2}{3}}-21 \,3^{\frac{1}{6}}\right) \sqrt{38}+57 \,\mathrm{I} \,3^{\frac{1}{6}}+19 \,3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(27+6 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \,3^{\frac{5}{6}} \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{2}+\frac{\left(\left(9 \,\mathrm{I}-2 \sqrt{38}\right) 3^{\frac{1}{6}}+\left(-2 \,\mathrm{I} \sqrt{38}+3\right) 3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}\right)^{-n}}{17100}\\+\\\frac{\left(\left(\left(-15 \,\mathrm{I} \,3^{\frac{1}{3}}-5 \,3^{\frac{5}{6}}\right) \sqrt{38}+285 \,\mathrm{I} \,3^{\frac{5}{6}}+285 \,3^{\frac{1}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}+2850+\left(\left(21 \,\mathrm{I} \,3^{\frac{2}{3}}-21 \,3^{\frac{1}{6}}\right) \sqrt{38}-57 \,\mathrm{I} \,3^{\frac{1}{6}}+19 \,3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(27+6 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \,3^{\frac{5}{6}} \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{2}+\frac{\left(\left(-9 \,\mathrm{I}-2 \sqrt{38}\right) 3^{\frac{1}{6}}+\left(2 \,\mathrm{I} \sqrt{38}+3\right) 3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}\right)^{-n}}{17100}\\-\\\frac{\left(\left(-\frac{3^{\frac{5}{6}} \sqrt{38}}{57}+3^{\frac{1}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}-5+\left(-\frac{7 \sqrt{38}\, 3^{\frac{1}{6}}}{95}+\frac{3^{\frac{2}{3}}}{15}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(2 \sqrt{38}\, 3^{\frac{1}{6}}-3 \,3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}}{150}-\frac{\left(27+6 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{2}\right)^{-n}}{30} & \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) = 2*a(n)-3*a(n+1)+4*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) = 2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15413/
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[20,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[18,x]+F[4,x]
F[18,x] = F[16,x]
F[19,x] = F[13,x]*F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[15,x]+F[22,x]+F[51,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[33,x]
F[24,x] = F[25,x]+F[4,x]
F[25,x] = F[15,x]+F[26,x]+F[30,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[17,x]+F[28,x]
F[28,x] = F[29,x]+F[31,x]
F[29,x] = F[15,x]+F[16,x]+F[30,x]
F[30,x] = F[11,x]*F[4,x]
F[31,x] = F[32,x]
F[32,x] = F[28,x]*F[4,x]
F[33,x] = F[34,x]+F[36,x]
F[34,x] = F[15,x]+F[22,x]+F[35,x]
F[35,x] = F[2,x]*F[4,x]
F[36,x] = F[15,x]+F[37,x]+F[38,x]+F[45,x]
F[37,x] = 0
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[34,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[44,x]+F[49,x]
F[44,x] = F[15,x]+F[37,x]+F[42,x]+F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[15,x]+F[22,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[2,x]+F[46,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[43,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[48,x]+F[53,x]
F[53,x] = F[46,x]+F[54,x]
F[54,x] = F[15,x]+F[37,x]+F[42,x]+F[55,x]
F[55,x] = F[4,x]*F[53,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_{20}\! \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_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{51}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{33}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{31}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{37}\! \left(x \right) = 0
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_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{42}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \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_20(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_18(x) + F_4(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_13(x)*F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_15(x) + F_22(x) + F_51(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_33(x))
Eq(F_24(x), F_25(x) + F_4(x))
Eq(F_25(x), F_15(x) + F_26(x) + F_30(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_17(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_31(x))
Eq(F_29(x), F_15(x) + F_16(x) + F_30(x))
Eq(F_30(x), F_11(x)*F_4(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_28(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_36(x))
Eq(F_34(x), F_15(x) + F_22(x) + F_35(x))
Eq(F_35(x), F_2(x)*F_4(x))
Eq(F_36(x), F_15(x) + F_37(x) + F_38(x) + F_45(x))
Eq(F_37(x), 0)
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_34(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_44(x) + F_49(x))
Eq(F_44(x), F_15(x) + F_37(x) + F_42(x) + F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_15(x) + F_22(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_2(x) + F_46(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_43(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_48(x) + F_53(x))
Eq(F_53(x), F_46(x) + F_54(x))
Eq(F_54(x), F_15(x) + F_37(x) + F_42(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_53(x))
Pack JSON:
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Specification JSON:
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