0231_0312_1302_1320_2301

Counting sequence:
1, 1, 2, 6, 19, 59, 180, 543, 1625, 4832, 14291, 42073, 123376, 360559, 1050589, 3053240, 8853115, 25618489, 74000016, 213410647, 614582389, 1767617352, 5078049315, 14573226785, 41783858880, 119700035743, 342647472925, 980164993464, 2802056422667, 8005811942761, 22861619638576, 65253195062311, 186169212310341, 530935317025192, 1513623548521907, 4313701560465265, 12289960571859104, 35005006602867311, 99677979735672493, 283770267686818200, 807684134555246107, 2298427881567963321, 6539459917748296720, 18602913142180293495, 52911974451585525333, 150475665716933887880, 427880787486481526851, 1216544523759727708481, 3458486343721223318400, 9831068968623691839807, 27943159821349197725757, 79417016807199328503672, 225692867732992613537323, 641343948194050797501193, 1822365728150824471862000, 5177900748521479899600327, 14711184284863730152203877, 41794506629421270517800296, 118732657651531976106324435, 337289031959196226629056529, 958113121616318520193447520, 2721543448010984113700416271, 7730346080608528811909679821, 21956795132877806799864821080, 62362911471752579094520321339, 177121884257421652581697260121, 503045633263894925970906171856, 1428672547795081801234026713879, 4057410461261178436257859186549, 11522768295431255732674939304776, 32723293522766123180007782613603, 92928747728309698496823520676833, 263898314011884777845670709354816, 749405908079928587259601526347743, 2128100078830234740048904050058845, 6043128876429218459542955529345848, 17160381495667024825179087276485451, 48729018536541168967998965992810153, 138370775133032202793751640329750704, 392913889125400278838170884213879655, 1115698823070183734464694730055815557, 3168060176379983411261856119315136808, 8995741532466725722018739868795040243, 25543349203082730900249924632733340849, 72529761092033443643173699275159768096, 205945534658569276550615322074922569647, 584771794728998259727189323786621936365, 1660422164598018244748529611595802368280, 4714643353133235301157327430660518914139, 13386823154486679744245890623455280832505, 38010597162933242451349538457966212691856, 107927088814135191000067604890528275334839, 306446697507705134201880896587053782162581, 870118176807835729696164433840555852402440, 2470588653357937199417389079600734060826755, 7014902429095559981622807348493306685907585, 19917824281676249655776323139281191839586048, 56553736064489150463801657938163891351855743, 160575733206684188232843082543920269632901245, 455929609436406528131853914039985588823887608, 1294538612743447828718557221224379277317250155

Generating function in Maple syntax:
-(3*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{3 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:
(-3*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)+3*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)+3 x^{4}-9 x^{3}+12 x^{2}-6 x +1 = 0

Explicit closed form in Maple syntax:
-1/9216*((13/2*((I-1/11*11^(1/2))*3^(1/2)-35/143*I*11^(1/2)+9/13)*2^(2/3)*(13+3*11^(1/2)*3^(1/2))^(2/3)+8*((I+2/11*11^(1/2))*3^(1/2)-13/11*I*11^(1/2))*2^(1/3)*(13+3*11^(1/2)*3^(1/2))^(1/3)-16/11*I*11^(1/2)-144)*(13/384*((I-3/13*11^(1/2))*3^(1/2)-9/13*I*11^(1/2)+1)*2^(2/3)*(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)+(((12/11*11^(1/2)+I)*3^(1/2)+1/11*I*11^(1/2)-12)*2^(2/3)*(13+3*11^(1/2)*3^(1/2))^(2/3)+4*2^(1/3)*((I-15/11*11^(1/2))*3^(1/2)-7/11*I*11^(1/2)+3)*(13+3*11^(1/2)*3^(1/2))^(1/3)+16/11*I*11^(1/2)-144)*(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)-3/2*5^(1/2)*(((I+1/15*11^(1/2))*3^(1/2)-1/5*I*11^(1/2)-1)*2^(2/3)*(13+3*11^(1/2)*3^(1/2))^(2/3)-256/5+8/5*2^(1/3)*((I-1/3*11^(1/2))*3^(1/2)-I*11^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(1/3))*(3/2-1/2*5^(1/2))^(-n)+3/2*5^(1/2)*(((I+1/15*11^(1/2))*3^(1/2)-1/5*I*11^(1/2)-1)*2^(2/3)*(13+3*11^(1/2)*3^(1/2))^(2/3)-256/5+8/5*2^(1/3)*((I-1/3*11^(1/2))*3^(1/2)-I*11^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(1/3))*(3/2+1/2*5^(1/2))^(-n)-96*(-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))*(((I+1/11*11^(1/2))*3^(1/2)-3/11*I*11^(1/2)-1)*2^(2/3)*(13+3*11^(1/2)*3^(1/2))^(2/3)+32+2^(1/3)*((I-5/11*11^(1/2))*3^(1/2)-15/11*I*11^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(1/3))

Explicit closed form in latex syntax:
-\frac{\left(\left(\frac{13 \left(\left(\mathrm{I}-\frac{\sqrt{11}}{11}\right) \sqrt{3}-\frac{35 \,\mathrm{I} \sqrt{11}}{143}+\frac{9}{13}\right) 2^{\frac{2}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{2}+8 \left(\left(\mathrm{I}+\frac{2 \sqrt{11}}{11}\right) \sqrt{3}-\frac{13 \,\mathrm{I} \sqrt{11}}{11}\right) 2^{\frac{1}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{16 \,\mathrm{I} \sqrt{11}}{11}-144\right) \left(\frac{13 \left(\left(\mathrm{I}-\frac{3 \sqrt{11}}{13}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{11}}{13}+1\right) 2^{\frac{2}{3}} \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}+\left(\left(\left(\frac{12 \sqrt{11}}{11}+\mathrm{I}\right) \sqrt{3}+\frac{\mathrm{I} \sqrt{11}}{11}-12\right) 2^{\frac{2}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+4 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{15 \sqrt{11}}{11}\right) \sqrt{3}-\frac{7 \,\mathrm{I} \sqrt{11}}{11}+3\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{16 \,\mathrm{I} \sqrt{11}}{11}-144\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}-\frac{3 \sqrt{5}\, \left(\left(\left(\mathrm{I}+\frac{\sqrt{11}}{15}\right) \sqrt{3}-\frac{\mathrm{I} \sqrt{11}}{5}-1\right) 2^{\frac{2}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{256}{5}+\frac{8 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{\sqrt{11}}{3}\right) \sqrt{3}-\mathrm{I} \sqrt{11}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{2}+\frac{3 \sqrt{5}\, \left(\left(\left(\mathrm{I}+\frac{\sqrt{11}}{15}\right) \sqrt{3}-\frac{\mathrm{I} \sqrt{11}}{5}-1\right) 2^{\frac{2}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{256}{5}+\frac{8 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{\sqrt{11}}{3}\right) \sqrt{3}-\mathrm{I} \sqrt{11}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{5}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{2}-96 \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}\right) \left(\left(\left(\mathrm{I}+\frac{\sqrt{11}}{11}\right) \sqrt{3}-\frac{3 \,\mathrm{I} \sqrt{11}}{11}-1\right) 2^{\frac{2}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+32+2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{5 \sqrt{11}}{11}\right) \sqrt{3}-\frac{15 \,\mathrm{I} \sqrt{11}}{11}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right)}{9216}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = 2*a(n)-10*a(n+1)+18*a(n+2)-17*a(n+3)+7*a(n+4), n >= 5

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(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 5

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21988/
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[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[13,x]*F[9,x]
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]*F[7,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[11,x]*F[13,x]
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]*F[4,x]
F[16,x] = F[11,x]+F[17,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[12,x]
F[19,x] = F[20,x]+F[21,x]+F[25,x]
F[20,x] = 0
F[21,x] = F[13,x]*F[22,x]
F[22,x] = F[13,x]+F[23,x]
F[23,x] = F[20,x]+F[21,x]+F[24,x]
F[24,x] = F[13,x]*F[18,x]
F[25,x] = F[13,x]*F[17,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_{14}\! \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_{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_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{7}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{11}\! \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_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \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_{12}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \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_{13}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{25}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \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_14(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_8(x))
Eq(F_8(x), F_13(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_11(x)*F_7(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_11(x)*F_13(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x)*F_4(x))
Eq(F_16(x), F_11(x) + F_17(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_12(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_25(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_13(x)*F_22(x))
Eq(F_22(x), F_13(x) + F_23(x))
Eq(F_23(x), F_20(x) + F_21(x) + F_24(x))
Eq(F_24(x), F_13(x)*F_18(x))
Eq(F_25(x), F_13(x)*F_17(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, 0, 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, 0, 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, 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