0231_1203_2301_3012
Counting sequence:
1, 1, 2, 6, 20, 63, 181, 481, 1210, 2941, 7015, 16599, 39233, 92989, 221417, 529918, 1274470, 3078480, 7463632, 18151100, 44254523, 108121999, 264610056, 648482954, 1591034947, 3907142589, 9602036265, 23611942994, 58091859751, 142979553028, 352026230268, 866945770545, 2135516734364, 5261269154898, 12964035128247, 31947754812232, 78737470973074, 194068855207092, 478362610606833, 1179181139989842, 2906843194213593, 7166005152420568, 17666248546391859, 43553298982204237, 107375570606872182, 264725737061784519, 652667399972398130, 1609132442028127935, 3967299610690492611, 9781397695048817594, 24116208577848966226, 59459181718268142073, 146598751169065462635, 361445470018652258372, 891161165784456451883, 2197204559352503589965, 5417330804145176551719, 13356747888374535515591, 32931879153235728697934, 81195625468317687225872, 200193061541839556096076, 493589179824724666379249, 1216977135073149334898809, 3000539519951483456871587, 7398035332228342718834786, 18240365911003364734474631, 44972887095982121997804779, 110883787932108545048670609, 273391740194557369562666329, 674066534831680529315429279, 1661958507002355725231968669, 4097675984138882004322708720, 10103109825447170261654797214, 24909932540918096765866903738, 61417204266486698898503586404, 151428474392964856810339867302, 373357654972714291567846443731, 920539822946561115958019452593, 2269656364543703850509524591324, 5595999157244425245122875199644, 13797333972867410172160622847691, 34018308594415716207242080787815, 83874560776133270088904637356765, 206798699633205090606631640670732, 509876915599697882145212893911695, 1257137836889759328897373859527606, 3099562849117350062849259066333440, 7642192920134071025677672028067331, 18842370868247559319820247145298068, 46457207264929427428767871242752844, 114543553169397116371658196567109771, 282415288338713986659564034001952866, 696315008064718607780888491960015034, 1716814247597615768369947837689557350, 4232927810131251837412519893321145701, 10436585016827564016859963661474808836, 25732143739639803556921014293924363089, 63444433265201068398489629751460126722, 156426769333859785345669000927041379383, 385681342069896885032462222645489105495, 950924821088979168539915424826889819758
Generating function in Maple syntax:
-(8*x^9-63*x^8+186*x^7-322*x^6+368*x^5-289*x^4+155*x^3-54*x^2+11*x-1)/(2*x-1)/(3*x^3-5*x^2+4*x-1)/(-1+x)^6
Generating function in latex syntax:
-\frac{8 x^{9}-63 x^{8}+186 x^{7}-322 x^{6}+368 x^{5}-289 x^{4}+155 x^{3}-54 x^{2}+11 x -1}{\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(-1+x \right)^{6}}
Generating function in sympy syntax:
(-8*x**9 + 63*x**8 - 186*x**7 + 322*x**6 - 368*x**5 + 289*x**4 - 155*x**3 + 54*x**2 - 11*x + 1)/((x - 1)**6*(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)*(-1+x)^6*F(x)+8*x^9-63*x^8+186*x^7-322*x^6+368*x^5-289*x^4+155*x^3-54*x^2+11*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) \left(-1+x \right)^{6} F \! \left(x \right)+8 x^{9}-63 x^{8}+186 x^{7}-322 x^{6}+368 x^{5}-289 x^{4}+155 x^{3}-54 x^{2}+11 x -1 = 0
Explicit closed form in Maple syntax:
1/450120*(-3410*((-15/31*I+5/31*3^(1/2))*31^(1/2)+I*3^(1/2)-1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)+150040-1240*2^(1/3)*((3/62*I+1/62*3^(1/2))*31^(1/2)+I*3^(1/2)+1)*(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)+1/450120*(3410*((-15/31*I-5/31*3^(1/2))*31^(1/2)+I*3^(1/2)+1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)+150040+1240*2^(1/3)*((3/62*I-1/62*3^(1/2))*31^(1/2)+I*3^(1/2)-1)*(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)+1/450120*((1100*2^(2/3)*31^(1/2)*3^(1/2)-6820*2^(2/3))*(47+9*31^(1/2)*3^(1/2))^(1/3)+150040+(40*31^(1/2)*3^(1/2)*2^(1/3)+2480*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)+1/120*n^5-1/12*n^4-1/24*n^3-5/12*n^2-37/15*n+2^(n+1)-2
Explicit closed form in latex syntax:
\frac{\left(-3410 \left(\left(-\frac{15 \,\mathrm{I}}{31}+\frac{5 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+150040-1240 \,2^{\frac{1}{3}} \left(\left(\frac{3 \,\mathrm{I}}{62}+\frac{\sqrt{3}}{62}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) \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}}{450120}+\frac{\left(3410 \left(\left(-\frac{15 \,\mathrm{I}}{31}-\frac{5 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+150040+1240 \,2^{\frac{1}{3}} \left(\left(\frac{3 \,\mathrm{I}}{62}-\frac{\sqrt{3}}{62}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) \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}}{450120}+\frac{\left(\left(1100 \,2^{\frac{2}{3}} \sqrt{31}\, \sqrt{3}-6820 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+150040+\left(40 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}+2480 \,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}}{450120}+\frac{n^{5}}{120}-\frac{n^{4}}{12}-\frac{n^{3}}{24}-\frac{5 n^{2}}{12}-\frac{37 n}{15}+2^{n +1}-2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 181
a(7) = 481
a(8) = 1210
a(9) = 2941
a(n+4) = 1/120*n^5-1/8*n^4+3/8*n^3-11/8*n^2-6*a(n)+13*a(n+1)-13*a(n+2)+6*a(n+3)-53/60*n+3, n >= 10
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(6\right) = 181
a \! \left(7\right) = 481
a \! \left(8\right) = 1210
a \! \left(9\right) = 2941
a \! \left(n +4\right) = \frac{n^{5}}{120}-\frac{n^{4}}{8}+\frac{3 n^{3}}{8}-\frac{11 n^{2}}{8}-6 a \! \left(n \right)+13 a \! \left(n +1\right)-13 a \! \left(n +2\right)+6 a \! \left(n +3\right)-\frac{53 n}{60}+3, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21458/
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[13,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[29,x]+F[8,x]
F[8,x] = F[14,x]+F[9,x]
F[9,x] = F[10,x]^2
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[10,x]^2*F[13,x]*F[16,x]
F[16,x] = F[17,x]+F[24,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[23,x]
F[20,x] = 0
F[21,x] = F[13,x]*F[22,x]
F[22,x] = F[11,x]+F[19,x]
F[23,x] = F[13,x]*F[18,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[20,x]+F[26,x]+F[28,x]
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[11,x]+F[25,x]
F[28,x] = F[13,x]*F[24,x]
F[29,x] = F[30,x]
F[30,x] = F[10,x]^2*F[13,x]*F[31,x]
F[31,x] = F[32,x]+F[42,x]
F[32,x] = F[10,x]*F[33,x]*F[36,x]
F[33,x] = F[1,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[13,x]*F[33,x]*F[36,x]
F[36,x] = F[1,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[13,x]*F[39,x]
F[39,x] = F[1,x]+F[38,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[10,x]^2*F[13,x]
F[42,x] = F[10,x]*F[11,x]*F[43,x]
F[43,x] = F[11,x]+F[33,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{29}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{14}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10} \left(x \right)^{2}
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_{13}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{24}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \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_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{24}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{13}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{42}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right) F_{33}\! \left(x \right) F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{1}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{13}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right)+F_{33}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_29(x) + F_8(x))
Eq(F_8(x), F_14(x) + F_9(x))
Eq(F_9(x), F_10(x)**2)
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_13(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)**2*F_13(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_24(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_23(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_13(x)*F_22(x))
Eq(F_22(x), F_11(x) + F_19(x))
Eq(F_23(x), F_13(x)*F_18(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_20(x) + F_26(x) + F_28(x))
Eq(F_26(x), F_13(x)*F_27(x))
Eq(F_27(x), F_11(x) + F_25(x))
Eq(F_28(x), F_13(x)*F_24(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_10(x)**2*F_13(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_42(x))
Eq(F_32(x), F_10(x)*F_33(x)*F_36(x))
Eq(F_33(x), F_1(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_13(x)*F_33(x)*F_36(x))
Eq(F_36(x), F_1(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_13(x)*F_39(x))
Eq(F_39(x), F_1(x) + F_38(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_10(x)**2*F_13(x))
Eq(F_42(x), F_10(x)*F_11(x)*F_43(x))
Eq(F_43(x), F_11(x) + F_33(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "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, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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, 2, 3, 1], "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, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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