0231_0312_1320_2301
Counting sequence:
1, 1, 2, 6, 20, 65, 205, 634, 1935, 5847, 17526, 52183, 154503, 455278, 1336123, 3907411, 11392126, 33125335, 96093927, 278183198, 803835219, 2318953755, 6680086526, 19217781663, 55222182495, 158512679550, 454568766619, 1302447455315, 3728905124958, 10668284502983, 30501996285175, 87158092674846, 248917676590691, 710547059015691, 2027391185369534, 5782379538774319, 16485993101863375, 46986905653556286, 133876470805490411, 381336762344512643, 1085926462541179550, 3091639682098533207, 8799998736943116135, 25043178206833834718, 71255192096854696243, 202707748530880800955, 576576473787410700542, 1639765264635833705023, 4662830707065829150143, 13257614877709925058302, 37690607646483041142587, 107141253632713971060403, 304536736696178124956894, 865536295390836514955239, 2459781262580268398197463, 6989981755914690654048414, 19862159284876975770728195, 56435198389517911544190059, 160342612455160213527758398, 455538526658116192572096079, 1294137216543471749429015727, 3676344921483785651876683710, 10443219521221004976952044171, 29664442839434328747817525219, 84260281489872873251172267294, 239329218758953700610535931767, 679760211725412497998771358023, 1930651840641485016609118313566, 5483291293599689186258274257875, 15572869950740212725827778135835, 44226927115797285634601272494462, 125601945949827763285691975374175, 356695910538850394447157695706783, 1012960863789029177810401960644222, 2876607063203154847319231285556443, 8168875542441211946918906981912595, 23197349057348528006692459304205150, 65873269500783289211762255361909255, 187057563390046373295532796102027959, 531174103951276390814684891789418014, 1508323261553642739317370526227591075, 4283001207831499414587303217998852299, 12161817784632647213848348182035980990, 34533896754525938595111168714677892975, 98059512276259682381383384353415840655, 278439695675056761734825798473501020990, 790624127179786153062955216786277824043, 2244949952779218313290133871033205575491, 6374426700295640355563951278283931776926, 18099794061860873665563085826547849959319, 51393041797578411246416938162690033667879, 145926222863302941983721802819868058244574, 414343775500385361748040970810434228007539, 1176486304964941465004018078954792575877499, 3340501132935429695885379289689986566656510, 9484952560113066880249701008312947765703807, 26931316840980861968255818452182522928974719, 76467869259132987412938540390450249382915582, 217119805879867048245869730456691635155697275, 616480043556204257719011407194723114778135923, 1750401988310501271364129377122209160934238686
Generating function in Maple syntax:
(x^5-4*x^4+9*x^3-12*x^2+6*x-1)/(2*x^3-4*x^2+4*x-1)/(x^2-3*x+1)
Generating function in latex syntax:
\frac{x^{5}-4 x^{4}+9 x^{3}-12 x^{2}+6 x -1}{\left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(x**5 - 4*x**4 + 9*x**3 - 12*x**2 + 6*x - 1)/((x**2 - 3*x + 1)*(2*x**3 - 4*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x^3-4*x^2+4*x-1)*(x^2-3*x+1)*F(x)-x^5+4*x^4-9*x^3+12*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)-x^{5}+4 x^{4}-9 x^{3}+12 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/21120*(-165*((I+1/33*11^(1/2))*3^(1/2)+1/11*I*11^(1/2)+1)*2^(2/3)*(13+3*11^(1/2)*3^(1/2))^(2/3)+10560-660*2^(1/3)*((I+7/33*11^(1/2))*3^(1/2)-7/11*I*11^(1/2)-1)*(13+3*11^(1/2)*3^(1/2))^(1/3))*(13/384*2^(2/3)*((I-3/13*11^(1/2))*3^(1/2)-9/13*I*11^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(26+6*11^(1/2)*3^(1/2))^(1/3)+1/12*(26+6*11^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/21120*(660*2^(1/3)*((I-7/33*11^(1/2))*3^(1/2)-7/11*I*11^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(1/3)+10560+165*((I-1/33*11^(1/2))*3^(1/2)+1/11*I*11^(1/2)-1)*2^(2/3)*(13+3*11^(1/2)*3^(1/2))^(2/3))*(-13/384*2^(2/3)*((I+3/13*11^(1/2))*3^(1/2)-9/13*I*11^(1/2)-1)*(13+3*11^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(26+6*11^(1/2)*3^(1/2))^(1/3)+1/12*(26+6*11^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/21120*((280*3^(1/2)*2^(1/3)*11^(1/2)-1320*2^(1/3))*(13+3*11^(1/2)*3^(1/2))^(1/3)+10560+(10*3^(1/2)*2^(2/3)*11^(1/2)+330*2^(2/3))*(13+3*11^(1/2)*3^(1/2))^(2/3))*(1/192*2^(2/3)*(3*11^(1/2)*3^(1/2)-13)*(13+3*11^(1/2)*3^(1/2))^(2/3)-1/6*(26+6*11^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/21120*(-2112*5^(1/2)-10560)*(3/2-1/2*5^(1/2))^(-n)+1/10*(3/2+1/2*5^(1/2))^(-n)*(5^(1/2)-5))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-165 \left(\left(\mathrm{I}+\frac{\sqrt{11}}{33}\right) \sqrt{3}+\frac{\mathrm{I} \sqrt{11}}{11}+1\right) 2^{\frac{2}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+10560-660 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{7 \sqrt{11}}{33}\right) \sqrt{3}-\frac{7 \,\mathrm{I} \sqrt{11}}{11}-1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{13 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{11}}{13}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{11}}{13}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}\, \left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{21120}\\+\\\frac{\left(660 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{7 \sqrt{11}}{33}\right) \sqrt{3}-\frac{7 \,\mathrm{I} \sqrt{11}}{11}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+10560+165 \left(\left(\mathrm{I}-\frac{\sqrt{11}}{33}\right) \sqrt{3}+\frac{\mathrm{I} \sqrt{11}}{11}-1\right) 2^{\frac{2}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{13 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{11}}{13}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{11}}{13}-1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{\mathrm{I} \sqrt{3}\, \left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{21120}\\+\\\frac{\left(\left(280 \sqrt{3}\, 2^{\frac{1}{3}} \sqrt{11}-1320 \,2^{\frac{1}{3}}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+10560+\left(10 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{11}+330 \,2^{\frac{2}{3}}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{2}{3}} \left(3 \sqrt{11}\, \sqrt{3}-13\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}-\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{21120}\\+\frac{\left(-2112 \sqrt{5}-10560\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{21120}+\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \left(\sqrt{5}-5\right)}{10} & \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) = 65
a(n+5) = 2*a(n)-10*a(n+1)+18*a(n+2)-17*a(n+3)+7*a(n+4), 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) = 65
a \! \left(n +5\right) = 2 a \! \left(n \right)-10 a \! \left(n +1\right)+18 a \! \left(n +2\right)-17 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21417/
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[22,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[22,x]*F[7,x]
F[7,x] = F[39,x]+F[8,x]
F[8,x] = F[4,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[19,x]
F[11,x] = F[12,x]+F[23,x]
F[12,x] = F[13,x]*F[19,x]
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[22,x]
F[15,x] = F[16,x]+F[18,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[13,x]
F[18,x] = F[13,x]*F[19,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[19,x]*F[22,x]
F[22,x] = x
F[23,x] = F[24,x]
F[24,x] = F[22,x]*F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[19,x]*F[22,x]*F[27,x]*F[37,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[22,x]*F[28,x]
F[31,x] = F[20,x]+F[32,x]
F[32,x] = F[33,x]+F[34,x]+F[36,x]
F[33,x] = 0
F[34,x] = F[22,x]*F[35,x]
F[35,x] = F[29,x]+F[32,x]
F[36,x] = F[22,x]*F[31,x]
F[37,x] = F[11,x]+F[38,x]
F[38,x] = F[2,x]*F[20,x]
F[39,x] = F[19,x]*F[40,x]
F[40,x] = F[13,x]+F[38,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_{22}\! \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_{22}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{39}\! \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_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{19}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{23}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{22}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{19}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = x
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right) F_{22}\! \left(x \right) F_{27}\! \left(x \right) F_{37}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{22}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{20}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = 0
F_{34}\! \left(x \right) = F_{22}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{36}\! \left(x \right) = F_{22}\! \left(x \right) F_{31}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right) F_{20}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right)+F_{38}\! \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_22(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_22(x)*F_7(x))
Eq(F_7(x), F_39(x) + F_8(x))
Eq(F_8(x), F_4(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_19(x))
Eq(F_11(x), F_12(x) + F_23(x))
Eq(F_12(x), F_13(x)*F_19(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_22(x))
Eq(F_15(x), F_16(x) + F_18(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_13(x))
Eq(F_18(x), F_13(x)*F_19(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_19(x)*F_22(x))
Eq(F_22(x), x)
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_22(x)*F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_19(x)*F_22(x)*F_27(x)*F_37(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_22(x)*F_28(x))
Eq(F_31(x), F_20(x) + F_32(x))
Eq(F_32(x), F_33(x) + F_34(x) + F_36(x))
Eq(F_33(x), 0)
Eq(F_34(x), F_22(x)*F_35(x))
Eq(F_35(x), F_29(x) + F_32(x))
Eq(F_36(x), F_22(x)*F_31(x))
Eq(F_37(x), F_11(x) + F_38(x))
Eq(F_38(x), F_2(x)*F_20(x))
Eq(F_39(x), F_19(x)*F_40(x))
Eq(F_40(x), F_13(x) + F_38(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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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