0123_0231_1320_2031_2130
Counting sequence:
1, 1, 2, 6, 19, 55, 146, 359, 830, 1830, 3890, 8039, 16254, 32307, 63355, 122916, 236426, 451595, 857673, 1621239, 3052599, 5728836, 10721631, 20018713, 37302945, 69391627, 128894046, 239116072, 443107975, 820349413, 1517508226, 2805128453, 5182079942, 9567969000, 17657518710, 32573155777, 60066566650, 110730745385, 204071887363, 376004114458, 692643072138, 1275690303269, 2349145032357, 4325257166681, 7962678789627, 14657446019538, 26978333277327, 49651774400711, 91373821293641, 168143512860749, 309394960017362, 569277729498466, 1047407489139107, 1927036900715819, 3545270128146442, 6522219248822843, 11998579017264982, 22072625864598690, 40604034340591506, 74692406902682803, 137396844997961246, 252738231811504375, 464900207172455651, 855152953012444584, 1572981784487179410, 2893343006193318903, 5321976197704894873, 9789107503918518995, 18005731677361741671, 33118926864063217784, 60917182499966473431, 112047368981092410125, 206092422739445989625, 379071446554529650695, 697234655003416624030, 1282436413359765518900, 2358803820708433520743, 4338574083924710539721, 7979974818636726072170, 14677612418766781386249, 26996581517468945624302, 49654848646510091837076, 91330142670524185097414, 167983352813919227426749, 308971224198147875337194, 568289379729201663557725, 1045251496855073512840539, 1922524300942838173791182, 3536084917801333218598306, 6503892656033879895522497, 11962553555486906146434661, 22002614750465609109185253, 40469196263838739858102651, 74434583492787089669113334, 136906748731939617898545783, 251811101636409461243083243, 463153361233828361889986993, 851872712122713647927953017, 1566839602886179871036367874, 2881869604656486487725744414, 5300588275972373013536593107
Generating function in Maple syntax:
-(3*x^8+x^7-3*x^6+x^5+x^4+4*x^3-9*x^2+5*x-1)/(x^2+x-1)/(x^3+x^2+x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{3 x^{8}+x^{7}-3 x^{6}+x^{5}+x^{4}+4 x^{3}-9 x^{2}+5 x -1}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-3*x**8 - x**7 + 3*x**6 - x**5 - x**4 - 4*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**4*(x**2 + x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(x^3+x^2+x-1)*(x-1)^4*F(x)+3*x^8+x^7-3*x^6+x^5+x^4+4*x^3-9*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+3 x^{8}+x^{7}-3 x^{6}+x^{5}+x^{4}+4 x^{3}-9 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
1/5280*(((-8100*I-2700*3^(1/2))*11^(1/2)+10560*I*3^(1/2)+10560)*(17+3*11^(1/2)*3^(1/2))^(1/3)+31680+((-10665*I+3555*3^(1/2))*11^(1/2)+21945*I*3^(1/2)-21945)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/5280*(((8100*I-2700*3^(1/2))*11^(1/2)-10560*I*3^(1/2)+10560)*(17+3*11^(1/2)*3^(1/2))^(1/3)+31680+((10665*I+3555*3^(1/2))*11^(1/2)-21945*I*3^(1/2)-21945)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/5280*((5400*11^(1/2)*3^(1/2)-21120)*(17+3*11^(1/2)*3^(1/2))^(1/3)-7110*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2)+43890*(17+3*11^(1/2)*3^(1/2))^(2/3)+31680)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/5280*(40128*5^(1/2)-89760)*(-1/2*5^(1/2)-1/2)^(-n)+1/5280*(-40128*5^(1/2)-89760)*(1/2*5^(1/2)-1/2)^(-n)-1/6*n^3+5/4*n^2+53/12*n+17
Explicit closed form in latex syntax:
\frac{\left(\left(\left(-8100 \,\mathrm{I}-2700 \sqrt{3}\right) \sqrt{11}+10560 \,\mathrm{I} \sqrt{3}+10560\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+31680+\left(\left(-10665 \,\mathrm{I}+3555 \sqrt{3}\right) \sqrt{11}+21945 \,\mathrm{I} \sqrt{3}-21945\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{5280}+\frac{\left(\left(\left(8100 \,\mathrm{I}-2700 \sqrt{3}\right) \sqrt{11}-10560 \,\mathrm{I} \sqrt{3}+10560\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+31680+\left(\left(10665 \,\mathrm{I}+3555 \sqrt{3}\right) \sqrt{11}-21945 \,\mathrm{I} \sqrt{3}-21945\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{5280}+\frac{\left(\left(5400 \sqrt{11}\, \sqrt{3}-21120\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-7110 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}+43890 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+31680\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{5280}+\frac{\left(40128 \sqrt{5}-89760\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5280}+\frac{\left(-40128 \sqrt{5}-89760\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5280}-\frac{n^{3}}{6}+\frac{5 n^{2}}{4}+\frac{53 n}{12}+17
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 146
a(7) = 359
a(8) = 830
a(n+5) = -1/3*n^3+7/2*n^2-a(n)-2*a(n+1)-a(n+2)+a(n+3)+2*a(n+4)+53/6*n+16, n >= 9
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) = 19
a \! \left(5\right) = 55
a \! \left(6\right) = 146
a \! \left(7\right) = 359
a \! \left(8\right) = 830
a \! \left(n +5\right) = -\frac{n^{3}}{3}+\frac{7 n^{2}}{2}-a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+\frac{53 n}{6}+16, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17516/
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[19,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[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[11,x]
F[18,x] = F[14,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[47,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[15,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[22,x]
F[28,x] = 2*F[21,x]+F[29,x]+F[33,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[25,x]
F[32,x] = F[29,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[46,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[11,x]+F[14,x]
F[40,x] = F[36,x]+F[41,x]
F[41,x] = 2*F[21,x]+F[33,x]+F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[14,x]
F[45,x] = F[42,x]
F[46,x] = F[42,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[53,x]
F[49,x] = F[2,x]+F[50,x]
F[50,x] = F[21,x]+F[37,x]+F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[2,x]+F[36,x]
F[53,x] = F[27,x]+F[54,x]
F[54,x] = F[42,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_{19}\! \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_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{47}\! \left(x \right)
F_{21}\! \left(x \right) = 0
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_{26}\! \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_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)
F_{28}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)
F_{32}\! \left(x \right) = F_{29}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{46}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{40}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{33}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{14}\! \left(x \right)
F_{45}\! \left(x \right) = F_{42}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \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_{53}\! \left(x \right)
F_{49}\! \left(x \right) = F_{2}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{21}\! \left(x \right)+F_{37}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{53}\! \left(x \right) = F_{27}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{42}\! \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_19(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_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_47(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_15(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_22(x))
Eq(F_28(x), 2*F_21(x) + F_29(x) + F_33(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_25(x))
Eq(F_32(x), F_29(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_46(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_11(x) + F_14(x))
Eq(F_40(x), F_36(x) + F_41(x))
Eq(F_41(x), 2*F_21(x) + F_33(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_14(x))
Eq(F_45(x), F_42(x))
Eq(F_46(x), F_42(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_53(x))
Eq(F_49(x), F_2(x) + F_50(x))
Eq(F_50(x), F_21(x) + F_37(x) + F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_2(x) + F_36(x))
Eq(F_53(x), F_27(x) + F_54(x))
Eq(F_54(x), F_42(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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