0132_1032_1203_1230_2103
Counting sequence:
1, 1, 2, 6, 19, 54, 144, 373, 946, 2366, 5862, 14426, 35329, 86212, 209824, 509665, 1236148, 2994820, 7249434, 17537174, 42403759, 102491786, 247657872, 598305125, 1445183334, 3490355122, 8428989694, 20354029146, 49147522045, 118668338024, 286519631552, 691772773409, 1670185048872, 4032363347368, 9735317262578, 23503743738182, 56744176599755, 136994620183102, 330738057937808, 798479272136757, 1927712302506586, 4653932754495046, 11235630925215062, 27125292296283898, 65486395200205049, 158098413184193260, 381683829429871712, 921467190075138241, 2224620265960129116, 5370711504267859052, 12966050231179492234, 31302824761962931990, 75571723289397552231, 182446314627069965906, 440464432159477697936, 1063375325382569701093, 2567215352263413582734, 6197806525300677902330, 14962829314031391245038, 36123466829260159768378, 87209766055006313051125, 210543004608790709373520, 508295785700456956946304, 1227134595189546374186433, 2962565011356778604890448, 7152264682788043459606992, 17267094496274876050235618, 41686453894841974861420294, 100640002689689955476189059, 242966460016799205344384454, 586572924089100994699998864, 1416112310707122272513121909, 3418797550123856565560609282, 8253707419453280135772487630, 19926212404661493672846837958, 48106132257526300075179930650, 116138476972593647984799866865, 280383086299974259635827852436, 676904649751432417602810694432, 1634192386131869562940445719585, 3945289422620352925520101922564, 9524771232485677514462400954580, 22994831889639021905062051488762, 55514435015529318757721802767894, 134023701927623672904739854907167, 323561838883515589435188156957594, 781147379718085387560472309916048, 1885856598362781838693709760141285, 4552860576522913999738811599019190, 10991577751554400782885185851445442, 26536016079899866919151532947347934, 64063609911847341854335374053665434, 154663235904501700159325605900906477, 373390081722519250291279382654667960, 901443399352609065624827615163183680, 2176276880433381904073673978579393761, 5253997160229754769306150978672460408, 12684271200911986725635633951826103416, 30622539562088849921593791670175303634, 73929350325154285448323223502279625030, 178481240212516236681706275688589893179
Generating function in Maple syntax:
(x^8+2*x^7-3*x^6-5*x^5+x^4+2*x^3+4*x^2-4*x+1)/(x^2+2*x-1)/(x^3+x^2+x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{8}+2 x^{7}-3 x^{6}-5 x^{5}+x^{4}+2 x^{3}+4 x^{2}-4 x +1}{\left(x^{2}+2 x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**8 + 2*x**7 - 3*x**6 - 5*x**5 + x**4 + 2*x**3 + 4*x**2 - 4*x + 1)/((x - 1)**2*(x**2 + 2*x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+2*x-1)*(x^3+x^2+x-1)*(x-1)^2*F(x)+x^8+2*x^7-3*x^6-5*x^5+x^4+2*x^3+4*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+2 x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}+2 x^{7}-3 x^{6}-5 x^{5}+x^{4}+2 x^{3}+4 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/528*(((-11*I+9*11^(1/2))*3^(1/2)+27*I*11^(1/2)-11)*(17+3*11^(1/2)*3^(1/2))^(1/3)-220+((-176*I-30*11^(1/2))*3^(1/2)+90*I*11^(1/2)+176)*(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)+1/528*(((176*I-30*11^(1/2))*3^(1/2)-90*I*11^(1/2)+176)*(17+3*11^(1/2)*3^(1/2))^(2/3)-220+((11*I+9*11^(1/2))*3^(1/2)-27*I*11^(1/2)-11)*(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)+1/528*((60*11^(1/2)*3^(1/2)-352)*(17+3*11^(1/2)*3^(1/2))^(2/3)-18*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)+22*(17+3*11^(1/2)*3^(1/2))^(1/3)-220)*(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)+1/528*(-165*2^(1/2)+264)*(-1-2^(1/2))^(-n)+1/528*(165*2^(1/2)+264)*(2^(1/2)-1)^(-n)-1/4*n+1/4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(-11 \,\mathrm{I}+9 \sqrt{11}\right) \sqrt{3}+27 \,\mathrm{I} \sqrt{11}-11\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-220+\left(\left(-176 \,\mathrm{I}-30 \sqrt{11}\right) \sqrt{3}+90 \,\mathrm{I} \sqrt{11}+176\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}}{528}\\+\\\frac{\left(\left(\left(176 \,\mathrm{I}-30 \sqrt{11}\right) \sqrt{3}-90 \,\mathrm{I} \sqrt{11}+176\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-220+\left(\left(11 \,\mathrm{I}+9 \sqrt{11}\right) \sqrt{3}-27 \,\mathrm{I} \sqrt{11}-11\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}}{528}\\+\\\frac{\left(\left(60 \sqrt{11}\, \sqrt{3}-352\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-18 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}+22 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-220\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}}{528}\\+\frac{\left(-165 \sqrt{2}+264\right) \left(-1-\sqrt{2}\right)^{-n}}{528}+\frac{\left(165 \sqrt{2}+264\right) \left(\sqrt{2}-1\right)^{-n}}{528}-\frac{n}{4}\\+\frac{1}{4} & \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) = 54
a(6) = 144
a(7) = 373
a(8) = 946
a(n+2) = -1/2*a(n)-3/2*a(n+1)+3/2*a(n+4)-1/2*a(n+5)+1/2-1/2*n, 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) = 54
a \! \left(6\right) = 144
a \! \left(7\right) = 373
a \! \left(8\right) = 946
a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{2}-\frac{3 a \! \left(n +1\right)}{2}+\frac{3 a \! \left(n +4\right)}{2}-\frac{a \! \left(n +5\right)}{2}+\frac{1}{2}-\frac{n}{2}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19649/
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[24,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[22,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[7,x]
F[24,x] = F[2,x]+F[25,x]
F[25,x] = F[15,x]+F[26,x]+F[36,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[11,x]+F[14,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[2,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[25,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[32,x]+F[38,x]
F[38,x] = F[25,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_{24}\! \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_{22}\! \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_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \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_24(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_22(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_7(x))
Eq(F_24(x), F_2(x) + F_25(x))
Eq(F_25(x), F_15(x) + F_26(x) + F_36(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_11(x) + F_14(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_2(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_25(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_32(x) + F_38(x))
Eq(F_38(x), F_25(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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