0132_1032_1203_1302
Counting sequence:
1, 1, 2, 6, 20, 67, 222, 729, 2381, 7754, 25214, 81928, 266111, 864202, 2806273, 9112265, 29587890, 96072134, 311945596, 1012883067, 3288813894, 10678716665, 34673583029, 112584429050, 365559363742, 1186963827440, 3854047383799, 12514013318098, 40632746115137, 131933698050897, 428386026881762, 1390960692218566, 4516420997853156, 14664726862836147, 47616069019280366, 154608404917238105, 502010337333344477, 1630017326182574122, 5292633011830041854, 17185071439388109016, 55799576452142868079, 181180086636215728154, 588288045907609541313, 1910159286172896205721, 6202248242055039508882, 20138573539145429851398, 65389537529569808409420, 212318494655031502447595, 689393821636884091632854, 2238447677783801251888121, 7268193953752733212091077, 23599677523697742640759514, 76627671573756115183139230, 248808486680346853283230144, 807876081482895317958074855, 2623157158906151585274132642, 8517337792313732871905444225, 27655622089614650165840500129, 89797240853097005183043967938, 291569809520110857792082283142, 946721224571586908027988108212, 3073984506590366398570490810147, 9981164995042421402476607684606, 32408639160241495528988081670361, 105230190337543315628651253054317, 341680281721310026371804382934026, 1109428905741531733616102478014590, 3602293028717341825478337345523112, 11696571990858828752066876242462047, 37978530687731624541080459617186154, 123315514522222492018479377976681601, 400402960475581541597981008422753129, 1300100245932304380420109233870862194, 4221398981330255144531139496794473798, 13706796391533030981733023089169948956, 44505688315615928421898387698953764187, 144509062210211711363624939456543600998, 469217977548896017268467394178165368825, 1523541202798825746582485721850519002325, 4946907210910964340665249183640146399674, 16062506815310763241099798981275783353246, 52154632013886855908089985526388557370512, 169345026388480664434516382927928233132759, 549859846674392949965141800462585604793970, 1785383707054967619126700690271747182543809, 5797104481616961060277169031511150517087985, 18823080012429392536590782816387855094971042, 61118156879500163940417600583144119554871110, 198449408804541425542820475048904617721617284, 644361182758135536608187554562046226373830803, 2092227617842929293724432523099947545094321422, 6793420401470374674515857763176364984950239577, 22058097483052433146840881231844185880559100157, 71622192624282846980833692103262687038188384746, 232555798624570010697639030888424684892537562366, 755103934860205404750033883054391488114116603512, 2451807075177889806545168922044672018253546028111, 7960967565352786614923280342785435847475568081850, 25849099310556799640095422064713538615970680844097, 83931498235845168317482597188899750013253541757497, 272523862881237830361763733899002954217919191804754
Generating function in Maple syntax:
-(2*x-1)*(x^2-3*x+1)/(x-1)/(x^3-6*x^2+5*x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x^{2}-3 x +1\right)}{\left(x -1\right) \left(x^{3}-6 x^{2}+5 x -1\right)}
Generating function in sympy syntax:
(1 - 2*x)*(x**2 - 3*x + 1)/((x - 1)*(x**3 - 6*x**2 + 5*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^3-6*x^2+5*x-1)*F(x)+(2*x-1)*(x^2-3*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{3}-6 x^{2}+5 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x^{2}-3 x +1\right) = 0
Explicit closed form in Maple syntax:
1/3528*(42*I*3^(1/2)*(756+84*I*3^(1/2))^(1/3)-5*I*(756+84*I*3^(1/2))^(2/3)*3^(1/2)+42*(756+84*I*3^(1/2))^(1/3)+3*(756+84*I*3^(1/2))^(2/3))*(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)+1/3528*(I*3^(1/2)*(756+84*I*3^(1/2))^(2/3)-84*(756+84*I*3^(1/2))^(1/3)-9*(756+84*I*3^(1/2))^(2/3))*(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+1/3528*(-42*I*3^(1/2)*(756+84*I*3^(1/2))^(1/3)+4*I*(756+84*I*3^(1/2))^(2/3)*3^(1/2)+42*(756+84*I*3^(1/2))^(1/3)+6*(756+84*I*3^(1/2))^(2/3))*(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)
Explicit closed form in latex syntax:
\frac{\left(42 \,\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-5 \,\mathrm{I} \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}} \sqrt{3}+42 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+3 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}\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(\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-84 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-9 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}\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}+1+\frac{\left(-42 \,\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}} \sqrt{3}+42 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+6 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}\right) \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}}{3528}
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)+1, 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)+1, \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15499/
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[12,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]+F[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[10,x]
F[19,x] = F[14,x]
F[20,x] = F[12,x]*F[13,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[15,x]+F[23,x]+F[30,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[10,x]
F[26,x] = F[27,x]
F[27,x] = F[15,x]+F[23,x]+F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[2,x]+F[27,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[29,x]+F[32,x]
F[32,x] = F[22,x]+F[33,x]
F[33,x] = F[15,x]+F[34,x]+F[35,x]+F[39,x]
F[34,x] = 0
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[27,x]
F[38,x] = F[33,x]
F[39,x] = F[12,x]*F[32,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{22}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{34}\! \left(x \right) = 0
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{27}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_21(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_10(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_12(x)*F_13(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_30(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_10(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_15(x) + F_23(x) + F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_2(x) + F_27(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_29(x) + F_32(x))
Eq(F_32(x), F_22(x) + F_33(x))
Eq(F_33(x), F_15(x) + F_34(x) + F_35(x) + F_39(x))
Eq(F_34(x), 0)
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_27(x))
Eq(F_38(x), F_33(x))
Eq(F_39(x), F_12(x)*F_32(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, 3, 0, 2], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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