0132_0213_1203_2013_2103
Counting sequence:
1, 1, 2, 6, 19, 61, 197, 638, 2069, 6714, 21794, 70755, 229725, 745889, 2421850, 7863641, 25532994, 82904974, 269190547, 874055885, 2838041117, 9215060822, 29921113293, 97153242650, 315454594314, 1024274628963, 3325798821581, 10798800928441, 35063486341682, 113850424959345, 369670007675074, 1200310974960982, 3897385253713811, 12654730426478237, 41089651585069301, 133417260620190894, 433203124017016901, 1406601707949008442, 4567207056263131698, 14829628157638624739, 48151500158563341949, 156346938903248093009, 507655321722499038090, 1648346475351569974345, 5352147385326103736194, 17378313396243597872990, 56427029144612936922131, 183217412730929201108909, 594903202183211981884749, 1931638563675097639692422, 6271991018007145508262525, 20365026910168353685042842, 66124827006473993015331482, 214705964589366988474662883, 697145887818159337966368365, 2263618478561068751999196009, 7349923030485754720672432738, 23865050168880520429333356001, 77489331140049142574631379586, 251606277717448345017829194662, 816960451915827390070691051795, 2652653924414496022821111470589, 8613113188294964118699240236837, 27966602846903671846640223412446, 90806989029163070543826787111797, 294848441252688285757991835321146, 957366874935366677373638677347394, 3108550716189866742864069161921891, 10093400772589821935836505580846237, 32773066440745275899371751610047233, 106413478284377314624703793727080634, 345522393550004739663124964555966009, 1121904164484505086466773812027393474, 3642799939406874308979823066528251950, 11828097103677345765761597425032864915, 38405590046429988061395822538022206349, 124702167549492740021389351206442094205, 404905394572561117504333418229110097846, 1314719557612279135454726806444920130349, 4268867588175521712269022874056382158874, 13860925989776494866121086949841500110122, 45006143977441623192446024311314127727715, 146134031536724668477772622731578020136717, 474494219808750098100308054739846834427417, 1540673053800844102827350561621080179044498, 5002533981688444594012677102397897908794705, 16243125805445908451199590197002855304133954, 52741098190899718794749238932247969246946038, 171249270103511587860561330581620612318721171, 556042887177605534985510809511618101416063581, 1805459913457857866558935303000614802414936917, 5862291514327167710742172988514986015897024270, 19034740978066296889342763934082859366411563429, 61805415717826336048819717042324995539090608442, 200680924235061066618784174595642807512880685906, 651606867846413613690345334658346923696271342307, 2115754209539528004787841342760202768943163204541, 6869810764854219408415918880446575110051068654769, 22306135434880342627042891680330005860292635388906, 72427566794815924689506786461730781410099928220457, 235170832129651787093692501107120446998794897423618
Generating function in Maple syntax:
-(x^3+3*x^2-4*x+1)/(x^3-6*x^2+5*x-1)
Generating function in latex syntax:
-\frac{x^{3}+3 x^{2}-4 x +1}{x^{3}-6 x^{2}+5 x -1}
Generating function in sympy syntax:
(-x**3 - 3*x**2 + 4*x - 1)/(x**3 - 6*x**2 + 5*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^3-6*x^2+5*x-1)*F(x)+x^3+3*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-6 x^{2}+5 x -1\right) F \! \left(x \right)+x^{3}+3 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/3528*((12*I*3^(1/2)-10)*(756+84*I*3^(1/2))^(2/3)-98*I*(756+84*I*3^(1/2))^(1/3)*3^(1/2)-126*(756+84*I*3^(1/2))^(1/3)+2352)*(1/504*(5*I*3^(1/2)-3)*(756+84*I*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(756+84*I*3^(1/2))^(1/3)-1/12*(756+84*I*3^(1/2))^(1/3)+2)^(-n)+1/3528*((-11*I*3^(1/2)-13)*(756+84*I*3^(1/2))^(2/3)+112*I*(756+84*I*3^(1/2))^(1/3)*3^(1/2)-84*(756+84*I*3^(1/2))^(1/3)+2352)*(1/252*(-2*I*3^(1/2)-3)*(756+84*I*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(756+84*I*3^(1/2))^(1/3)-1/12*(756+84*I*3^(1/2))^(1/3)+2)^(-n)-1/252*(1/6*(756+84*I*3^(1/2))^(1/3)+2+1/56*(756+84*I*3^(1/2))^(2/3)-1/504*I*3^(1/2)*(756+84*I*3^(1/2))^(2/3))^(-n)*((1/14*I*3^(1/2)-23/14)*(756+84*I*3^(1/2))^(2/3)+I*(756+84*I*3^(1/2))^(1/3)*3^(1/2)-15*(756+84*I*3^(1/2))^(1/3)-168))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(12 \,\mathrm{I} \sqrt{3}-10\right) \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-98 \,\mathrm{I} \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-126 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+2352\right) \left(\frac{\left(5 \,\mathrm{I} \sqrt{3}-3\right) \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{504}-\frac{\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+2\right)^{-n}}{3528}\\+\\\frac{\left(\left(-11 \,\mathrm{I} \sqrt{3}-13\right) \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+112 \,\mathrm{I} \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-84 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+2352\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-3\right) \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{252}+\frac{\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+2\right)^{-n}}{3528}\\-\\\frac{\left(\frac{\left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{6}+2+\frac{\left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{56}-\frac{\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{504}\right)^{-n} \left(\left(\frac{\mathrm{I} \sqrt{3}}{14}-\frac{23}{14}\right) \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+\mathrm{I} \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-15 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-168\right)}{252} & \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) = a(n)-6*a(n+1)+5*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) = a \! \left(n \right)-6 a \! \left(n +1\right)+5 a \! \left(n +2\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18402/
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[16,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[13,x]*F[4,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[47,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[28,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[13,x]+F[25,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[4,x]
F[28,x] = F[29,x]+F[38,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[2,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[32,x]*F[4,x]
F[35,x] = F[29,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[35,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[17,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[41,x]
F[44,x] = F[38,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[44,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[32,x]+F[41,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_{16}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{47}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{38}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \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_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{17}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \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_16(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_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_47(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_28(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_13(x) + F_25(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_38(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_2(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_32(x)*F_4(x))
Eq(F_35(x), F_29(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_35(x)*F_4(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_44(x))
Eq(F_41(x), F_17(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_41(x))
Eq(F_44(x), F_38(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_44(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_32(x) + F_41(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": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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