0132_0321_2103_2130
Counting sequence:
1, 1, 2, 6, 20, 61, 168, 431, 1059, 2538, 5998, 14066, 32849, 76544, 178155, 414407, 963664, 2240572, 5209066, 12110023, 28152850, 65447949, 152148589, 353703180, 822260816, 1911525228, 4443757831, 10330484502, 24015463773, 55829180901, 129787100470, 301718404250, 701410193640, 1630580873881, 3790640639668, 8812170366003, 20485810693743, 47623731990150, 110711744950298, 257373581565742, 598320986788261, 1390927542185156, 3233514234546327, 7517008606058651, 17474924891270260, 40624271696241696, 94439973912245198, 219546303235524531, 510383233578327054, 1186497068176179553, 2758271040607411433, 6412202219048204696, 14906561644105973324, 34653551534828924664, 80559733535323034795, 187278659180417231242, 435370062005434551769, 1012112601190792230697, 2352876338741924822986, 5469773875849624562638, 12715681551255816276060, 29560373240810124529429, 68719530495768365602464, 159753526556940664028375, 371381891920095385413531, 863358153142173193790338, 2007064202143269474576486, 4665858192065557421566602, 10846804325052306509341593, 25215760793169074159472616, 58619531921468166881305955, 136273878503118658834319055, 316798332459588716899822840, 736466772293996999912141476, 1712077530465932224771103058, 3980097379269391391388854463, 9252603849170306724536504298, 21509694319438069615602912725, 50003972639242986789124589893, 116245133128023127860704454532, 270237148425021479619467102696, 628225151658261169926116995236, 1460446291252763678400121241327, 3395125718866790174967596842846, 7892709725753104338028665047925, 18348324030778496342550922706381, 42654678359696070526563034873278, 99160096743284323232615924262306, 230519257541239324987272625754224, 535892257496845399023149063618993, 1245798354151341870327517863618668, 2896129805001574137923528089380251, 6732684964198884072138697604530599, 15651593636744845810896554498458334, 36385540786838343426335796375702610, 84586120051224222729352977734730502, 196638872216740827146283894951253549, 457129917334612379406481525760011372, 1062698127621579706656229765112266831, 2470473420412255188302010138768040803, 5743153923328218531000052411839609820
Generating function in Maple syntax:
-(2*x^5-7*x^4+9*x^3-10*x^2+5*x-1)/(x^3-2*x^2+3*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{2 x^{5}-7 x^{4}+9 x^{3}-10 x^{2}+5 x -1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-2*x**5 + 7*x**4 - 9*x**3 + 10*x**2 - 5*x + 1)/((x - 1)**3*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x-1)^3*F(x)+2*x^5-7*x^4+9*x^3-10*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+2 x^{5}-7 x^{4}+9 x^{3}-10 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
1/600*(15*2^(2/3)*((I-1/3*3^(1/2))*23^(1/2)+I*3^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3)-24*2^(1/3)*((-1/4*I-1/12*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(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)+1/600*(-15*2^(2/3)*((I+1/3*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+24*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)*((-1/4*I+1/12*3^(1/2))*23^(1/2)+I*3^(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)+1/600*((10*2^(2/3)*23^(1/2)*3^(1/2)+30*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)-4*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)*(23^(1/2)*3^(1/2)-12))*(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)-n^2-n+1
Explicit closed form in latex syntax:
\frac{\left(15 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-24 \,2^{\frac{1}{3}} \left(\left(-\frac{\mathrm{I}}{4}-\frac{\sqrt{3}}{12}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{600}+\frac{\left(-15 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+24 \,2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\left(-\frac{\mathrm{I}}{4}+\frac{\sqrt{3}}{12}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-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}}{600}+\frac{\left(\left(10 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}+30 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-4 \,2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\sqrt{23}\, \sqrt{3}-12\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}}{600}-n^{2}-n +1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 61
a(n+3) = n^2+a(n)-2*a(n+1)+3*a(n+2)+3*n+1, 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) = 61
a \! \left(n +3\right) = n^{2}+a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right)+3 n +1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15462/
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[11,x]+F[18,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[36,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[15,x]+F[26,x]+F[28,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[17,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[13,x]+F[30,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = 2*F[15,x]+F[32,x]+F[35,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[18,x]+F[34,x]
F[34,x] = F[32,x]
F[35,x] = F[30,x]*F[4,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[45,x]
F[38,x] = F[2,x]+F[39,x]
F[39,x] = F[15,x]+F[40,x]+F[44,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[11,x]+F[43,x]
F[43,x] = F[15,x]+F[19,x]+F[26,x]
F[44,x] = F[38,x]*F[4,x]
F[45,x] = F[39,x]+F[46,x]
F[46,x] = F[15,x]+F[47,x]+F[54,x]+F[55,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[14,x]+F[50,x]
F[50,x] = F[15,x]+F[35,x]+F[51,x]+F[53,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[33,x]
F[53,x] = 0
F[54,x] = 0
F[55,x] = F[4,x]*F[45,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_{11}\! \left(x \right)+F_{18}\! \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_{36}\! \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_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \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_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{44}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{14}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{33}\! \left(x \right)
F_{53}\! \left(x \right) = 0
F_{54}\! \left(x \right) = 0
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \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_11(x) + F_18(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_36(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_15(x) + F_26(x) + F_28(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_17(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_13(x) + F_30(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), 2*F_15(x) + F_32(x) + F_35(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_18(x) + F_34(x))
Eq(F_34(x), F_32(x))
Eq(F_35(x), F_30(x)*F_4(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_45(x))
Eq(F_38(x), F_2(x) + F_39(x))
Eq(F_39(x), F_15(x) + F_40(x) + F_44(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_11(x) + F_43(x))
Eq(F_43(x), F_15(x) + F_19(x) + F_26(x))
Eq(F_44(x), F_38(x)*F_4(x))
Eq(F_45(x), F_39(x) + F_46(x))
Eq(F_46(x), F_15(x) + F_47(x) + F_54(x) + F_55(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_14(x) + F_50(x))
Eq(F_50(x), F_15(x) + F_35(x) + F_51(x) + F_53(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_33(x))
Eq(F_53(x), 0)
Eq(F_54(x), 0)
Eq(F_55(x), F_4(x)*F_45(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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