0123_1302_1320_2031_2130
Counting sequence:
1, 1, 2, 6, 19, 53, 137, 330, 752, 1643, 3474, 7160, 14463, 28751, 56425, 109592, 211060, 403653, 767550, 1452510, 2737683, 5142493, 9631997, 17996886, 33556116, 62455235, 116064066, 215402408, 399305531, 739483899, 1368288717, 2529888168, 4674569020, 8632465589, 15933552218, 29396937570, 54215937279, 99955761533, 184230954677, 339474308286, 625394999736, 1151905894999, 2121314812942, 3905960952400, 7191066517611, 13237572387699, 24365714822609, 44844698800580, 82529446051600, 151871664791453, 279459074804442, 514205255928546, 946094330970027, 1740662067433529, 3202423395512233, 5891544940830622, 10838457291875696, 19938617663237207, 36678638819066170, 67471924732325220, 124115411095902407, 228308415486718947, 419964422035650905, 772499360178407292, 1420951979981622244, 2613706656036666497, 4807628672318533302, 8843048878299651674, 16265616452739524875, 29918287819405219585, 55030179212576586173, 101219303362901037514, 186176216335194747508, 342439364729163989839, 629856996186063305482, 1158508354827717200052, 2130862605079043191395, 3919321623006217558575, 7208844139162470516317, 13259273590410617703084, 24387836131991684790156, 44856595864140038467857, 82504744368529985441138, 151750857149224618647294, 279114916948445196994519, 513374918817313265479617, 944247812832647099966869, 1736749168867183045693906, 3194390540703584913249784, 5875417682858634044284163, 10806606193071062490722422, 19876493377730160921136380, 36558645015398397416529651, 67241951309035040261667527, 123677424186737557993011969, 227478561718580374498179560, 418398812906336310973521088, 769556215711046960512358553, 1415435882927339701252380946, 2603394621035491745054242334, 4788392721756023234403299683
Generating function in Maple syntax:
(x^8-x^7-3*x^6+3*x^5-x^3-5*x^2+4*x-1)/(x^2+x-1)/(x^3+x^2+x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{8}-x^{7}-3 x^{6}+3 x^{5}-x^{3}-5 x^{2}+4 x -1}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**8 - x**7 - 3*x**6 + 3*x**5 - x**3 - 5*x**2 + 4*x - 1)/((x - 1)**3*(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)^3*F(x)-x^8+x^7+3*x^6-3*x^5+x^3+5*x^2-4*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)^{3} F \! \left(x \right)-x^{8}+x^{7}+3 x^{6}-3 x^{5}+x^{3}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/5280*(((-2440*11^(1/2)+9460*I)*3^(1/2)-7320*I*11^(1/2)+9460)*(17+3*11^(1/2)*3^(1/2))^(1/3)+28160+((3275*11^(1/2)+20185*I)*3^(1/2)-9825*I*11^(1/2)-20185)*(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*(((3275*11^(1/2)-20185*I)*3^(1/2)+9825*I*11^(1/2)-20185)*(17+3*11^(1/2)*3^(1/2))^(2/3)+28160+((-2440*11^(1/2)-9460*I)*3^(1/2)+7320*I*11^(1/2)+9460)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(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*((-6550*11^(1/2)*3^(1/2)+40370)*(17+3*11^(1/2)*3^(1/2))^(2/3)+4880*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-18920*(17+3*11^(1/2)*3^(1/2))^(1/3)+28160)*(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*(38016*5^(1/2)-84480)*(-1/2*5^(1/2)-1/2)^(-n)+1/5280*(-38016*5^(1/2)-84480)*(1/2*5^(1/2)-1/2)^(-n)+3/4*n^2+23/4*n+16)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(-2440 \sqrt{11}+9460 \,\mathrm{I}\right) \sqrt{3}-7320 \,\mathrm{I} \sqrt{11}+9460\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+28160+\left(\left(3275 \sqrt{11}+20185 \,\mathrm{I}\right) \sqrt{3}-9825 \,\mathrm{I} \sqrt{11}-20185\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(3275 \sqrt{11}-20185 \,\mathrm{I}\right) \sqrt{3}+9825 \,\mathrm{I} \sqrt{11}-20185\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+28160+\left(\left(-2440 \sqrt{11}-9460 \,\mathrm{I}\right) \sqrt{3}+7320 \,\mathrm{I} \sqrt{11}+9460\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{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(-6550 \sqrt{11}\, \sqrt{3}+40370\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+4880 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-18920 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+28160\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(38016 \sqrt{5}-84480\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5280}+\\\frac{\left(-38016 \sqrt{5}-84480\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5280}+\frac{3 n^{2}}{4}+\frac{23 n}{4}+16 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 137
a(7) = 330
a(8) = 752
a(n+5) = 3/2*n^2-a(n)-2*a(n+1)-a(n+2)+a(n+3)+2*a(n+4)+17/2*n+13, 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) = 53
a \! \left(6\right) = 137
a \! \left(7\right) = 330
a \! \left(8\right) = 752
a \! \left(n +5\right) = \frac{3 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{17 n}{2}+13, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18056/
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[23,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[16,x]+F[21,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[11,x]
F[19,x] = F[20,x]
F[20,x] = F[16,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[7,x]
F[23,x] = F[2,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[48,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[29,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[16,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[25,x]
F[31,x] = 2*F[15,x]+F[32,x]+F[36,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[28,x]
F[35,x] = F[32,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[11,x]+F[20,x]
F[42,x] = F[38,x]+F[43,x]
F[43,x] = 2*F[15,x]+F[36,x]+F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[20,x]
F[47,x] = F[44,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[2,x]+F[51,x]
F[51,x] = F[15,x]+F[39,x]+F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[2,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_{23}\! \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_{16}\! \left(x \right)+F_{21}\! \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_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{2}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{48}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{28}\! \left(x \right)
F_{35}\! \left(x \right) = F_{32}\! \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_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{11}\! \left(x \right)+F_{20}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{20}\! \left(x \right)
F_{47}\! \left(x \right) = F_{44}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{2}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{2}\! \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_23(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) + F_16(x) + F_21(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_11(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_16(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_7(x))
Eq(F_23(x), F_2(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_48(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_29(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_16(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_25(x))
Eq(F_31(x), 2*F_15(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_28(x))
Eq(F_35(x), F_32(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_11(x) + F_20(x))
Eq(F_42(x), F_38(x) + F_43(x))
Eq(F_43(x), 2*F_15(x) + F_36(x) + F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_20(x))
Eq(F_47(x), F_44(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_2(x) + F_51(x))
Eq(F_51(x), F_15(x) + F_39(x) + F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_2(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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