0123_0132_1203_1320_2013

Counting sequence:
1, 1, 2, 6, 19, 57, 161, 432, 1114, 2788, 6821, 16399, 38891, 91234, 212152, 489790, 1124031, 2566677, 5836101, 13222180, 29862966, 67265928, 151161865, 339002395, 758902159, 1696229110, 3785986516, 8439927042, 18794186947, 41810567969, 92933478153, 206405973656, 458112176146, 1016137152236, 2252638921773, 4991307307047, 11054586341107, 24473480753610, 54161607113584, 119824477166982, 265015788285063, 585979890078765, 1295357670114701, 2862877954094924, 6326029449429614, 13976003871296016, 30872060301331217, 68184499261380659, 150573700812634647, 332476858763755230, 734053010463200652, 1620509309087655114, 3577154651636303883, 7895680663130285369, 17426507334137179857, 38459442717488571712, 84873112893263245130, 187289826710975302900, 413273283320062443189, 911888053894634662975, 2012002683222737691899, 4439152147210523953202, 9793939343205654821928, 21607375359413991840974, 47668890845598396645711, 105161696993522226134597, 231990721264698000152789, 511770237211473509036660, 1128941979089427468378790, 2490354294789469385252248, 5493438057482245176225049, 12117736555437583614196811, 26729664328431968200381855, 58960440559880844750460870, 130053965366268599462062916, 286868290443107819818396562, 632758412210373789775031891, 1395693571315570789787682641, 3078500996131358620944873881, 6790251530587309474767003880, 14977178884718626625526139010, 33034823270025485644406049276, 72863827079552028308396897341, 160712691061650178331955523671, 354475921428980832487588276739, 781847102008823673642115811226, 1724469759221917486333271867104, 3803541168158055726588301453078, 8389180894895414969687057601815, 18503334462153707111444064839773, 40811215918747389320949786670877, 90013624384954032354533342020252, 198534606537189449306404171033822, 437888475603381642905544973045792, 965806668812228028109196976726561, 2130180130602666925411945501716579, 4698313109690758333503730730938407, 10362561633957830374665247947522478, 22855560890046499033839620414599932, 50409949872840099119377329595815258, 111183491345750714049808623210507163

Generating function in Maple syntax:
(x^6-2*x^5+4*x^4-6*x^3+9*x^2-5*x+1)/(2*x-1)/(x^3+2*x-1)/(x-1)^2

Generating function in latex syntax:
\frac{x^{6}-2 x^{5}+4 x^{4}-6 x^{3}+9 x^{2}-5 x +1}{\left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{2}}

Generating function in sympy syntax:
(x**6 - 2*x**5 + 4*x**4 - 6*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**2*(2*x - 1)*(x**3 + 2*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^3+2*x-1)*(x-1)^2*F(x)-x^6+2*x^5-4*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(x^{3}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)-x^{6}+2 x^{5}-4 x^{4}+6 x^{3}-9 x^{2}+5 x -1 = 0

Explicit closed form in Maple syntax:
2-2620/59*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+53/59*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-5215/59*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+2730/59*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+n+piecewise(n = 0,1/2,0)

Explicit closed form in latex syntax:
2-\frac{2620 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{59}+\frac{53 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{59}-\frac{5215 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{59}+\frac{2730 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{59}+n +\left(\left\{\begin{array}{cc}\frac{1}{2} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 57
a(6) = 161
a(n+4) = -2*a(n)+a(n+1)-4*a(n+2)+4*a(n+3)+2*n+3, n >= 7

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) = 19
a \! \left(5\right) = 57
a \! \left(6\right) = 161
a \! \left(n +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)+2 n +3, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16423/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[45,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[29,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[18,x]
F[29,x] = F[20,x]+F[30,x]
F[30,x] = 2*F[13,x]+F[31,x]+F[35,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[24,x]
F[34,x] = F[30,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[7,x]
F[42,x] = F[38,x]
F[43,x] = F[44,x]
F[44,x] = F[38,x]*F[4,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[2,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[2,x]*F[4,x]
F[50,x] = F[38,x]+F[51,x]
F[51,x] = F[44,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \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_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{45}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{24}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \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_{43}\! \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_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{2}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{50}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{44}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_45(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_29(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_18(x))
Eq(F_29(x), F_20(x) + F_30(x))
Eq(F_30(x), 2*F_13(x) + F_31(x) + F_35(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_24(x))
Eq(F_34(x), F_30(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_43(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_42(x))
Eq(F_41(x), F_7(x))
Eq(F_42(x), F_38(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_38(x)*F_4(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_2(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_2(x)*F_4(x))
Eq(F_50(x), F_38(x) + F_51(x))
Eq(F_51(x), F_44(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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