0213_0231_1320_2103
Counting sequence:
1, 1, 2, 6, 20, 63, 184, 509, 1361, 3563, 9198, 23508, 59627, 150354, 377389, 943835, 2353793, 5856836, 14547228, 36080799, 89386940, 221244201, 547202005, 1352582843, 3341723966, 8252927696, 20375519547, 50291971734, 124107792545, 306215310663, 755433076153, 1863448890980, 4596215639240, 11335796378711, 27956212087968, 68942029568805, 170009479357049, 419226470619147, 1033744703939902, 2548995159819308, 6285177049603739, 15497447547475274, 38211852494014629, 94217627532841971, 232307438594417777, 572784870776899572, 1412270565896824180, 3482111011811629919, 8585507402744252948, 21168409394911645649, 52192679891964829437, 128685687382743091995, 317285563418301645038, 782293826760222583080, 1928808492747285332299, 4755629408742281247278, 11725372414112061873593, 28909799616176088848351, 71279287670765672954217, 175744395945307999569780, 433310796477511032800864, 1068359573598959417712759, 2634118508650523036547160, 6494610573652461218009645, 16012932507379374066405473, 39481040757657291533143979, 97343348355590743950435150, 240007019438016769326952452, 591754522810689490138496651, 1459013168246654861737391106, 3597301375501639336171661149, 8869403745714185570282624267, 21868148643106178313219255137, 53917482036394039665448663652, 132937400299258496317485813708, 327766645208523666904236825951, 808132041975266995069622474156, 1992507175812790514797279703865, 4912668495113528844728749914277, 12112534359542225310487534543739, 29864316754492151603398359682142, 73632601234503112476742279686656, 181546425301988459984240451835035, 447615647414534916689046543689094, 1103628271082947072916412866112785, 2721074131627559825475544324454007, 6709002130264070161992061044205849, 16541522710017245688549378044664644, 40784302651331157319083073714537208, 100556603581430303116400055334534871, 247929469469916546529509922033630800, 611287768468106909680642502873827317, 1507173538350333349798928307867493577, 3716043721636943559130445964275197515, 9162170504533218354067900788531738910, 22589984033664735945013820180647318364, 55697214794234843237102438521343910587, 137325450556877669385535003244909745530, 338585680426655311391650576309212340821, 834807113443586712750197195365034280483, 2058276403589001870999066091226208480593
Generating function in Maple syntax:
(x^5+3*x^4-6*x^3+9*x^2-5*x+1)/(2*x-1)/(3*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
\frac{x^{5}+3 x^{4}-6 x^{3}+9 x^{2}-5 x +1}{\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(x**5 + 3*x**4 - 6*x**3 + 9*x**2 - 5*x + 1)/((2*x - 1)*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(3*x^3-5*x^2+4*x-1)*F(x)+x^5+3*x^4-6*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)+x^{5}+3 x^{4}-6 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/810216*(-5146*((I+102/2573*31^(1/2))*3^(1/2)+306/2573*I*31^(1/2)+1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)+510136-18755*2^(2/3)*((I+21/155*31^(1/2))*3^(1/2)-63/155*I*31^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(1/3))*(47/8712*2^(1/3)*((I-9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/810216*(18755*((I-21/155*31^(1/2))*3^(1/2)-63/155*I*31^(1/2)+1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)+510136+5146*((I-102/2573*31^(1/2))*3^(1/2)+306/2573*I*31^(1/2)-1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3))*(-47/8712*((I+9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)-1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)+1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/810216*((5082*2^(2/3)*31^(1/2)*3^(1/2)-37510*2^(2/3))*(47+9*31^(1/2)*3^(1/2))^(1/3)+510136+(408*31^(1/2)*3^(1/2)*2^(1/3)+10292*2^(1/3))*(47+9*31^(1/2)*3^(1/2))^(2/3))*(1/4356*2^(1/3)*(9*31^(1/2)*3^(1/2)-47)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/18*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)-7/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(-5146 \left(\left(\mathrm{I}+\frac{102 \sqrt{31}}{2573}\right) \sqrt{3}+\frac{306 \,\mathrm{I} \sqrt{31}}{2573}+1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+510136-18755 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{21 \sqrt{31}}{155}\right) \sqrt{3}-\frac{63 \,\mathrm{I} \sqrt{31}}{155}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{810216}\\+\\\frac{\left(18755 \left(\left(\mathrm{I}-\frac{21 \sqrt{31}}{155}\right) \sqrt{3}-\frac{63 \,\mathrm{I} \sqrt{31}}{155}+1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+510136+5146 \left(\left(\mathrm{I}-\frac{102 \sqrt{31}}{2573}\right) \sqrt{3}+\frac{306 \,\mathrm{I} \sqrt{31}}{2573}-1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{47 \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{810216}\\+\\\frac{\left(\left(5082 \,2^{\frac{2}{3}} \sqrt{31}\, \sqrt{3}-37510 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+510136+\left(408 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}+10292 \,2^{\frac{1}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{810216}\\-\frac{7 \,2^{n}}{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) = 20
a(5) = 63
a(n+4) = -6*a(n)+13*a(n+1)-13*a(n+2)+6*a(n+3), n >= 6
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) = 20
a \! \left(5\right) = 63
a \! \left(n +4\right) = -6 a \! \left(n \right)+13 a \! \left(n +1\right)-13 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15707/
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[11,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[6,x]
F[10,x] = F[7,x]
F[11,x] = F[12,x]+F[2,x]
F[12,x] = F[13,x]+F[14,x]+F[39,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[27,x]
F[16,x] = F[17,x]+F[7,x]
F[17,x] = F[13,x]+F[18,x]+F[20,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[16,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[10,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[23,x]+F[7,x]
F[27,x] = F[28,x]+F[32,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[2,x]+F[28,x]
F[32,x] = 2*F[13,x]+F[33,x]+F[35,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[27,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[26,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[11,x]+F[41,x]
F[41,x] = F[37,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_{11}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{7}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{2}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{39}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right)+F_{18}\! \left(x \right)+F_{20}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \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_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{28}\! \left(x \right)
F_{32}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{27}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \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_{26}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{11}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \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_11(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_6(x))
Eq(F_10(x), F_7(x))
Eq(F_11(x), F_12(x) + F_2(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_39(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_27(x))
Eq(F_16(x), F_17(x) + F_7(x))
Eq(F_17(x), F_13(x) + F_18(x) + F_20(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_16(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_10(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_23(x) + F_7(x))
Eq(F_27(x), F_28(x) + F_32(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_2(x) + F_28(x))
Eq(F_32(x), 2*F_13(x) + F_33(x) + F_35(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_27(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_26(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_11(x) + F_41(x))
Eq(F_41(x), F_37(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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