0231_1203_1302_2130

Counting sequence:
1, 1, 2, 6, 20, 67, 224, 751, 2534, 8621, 29597, 102558, 358639, 1265163, 4499978, 16128348, 58212018, 211450163, 772532618, 2837249058, 10469575053, 38798282948, 144334490018, 538820833923, 2017884062622, 7578846105224, 28540206246751, 107736571024378, 407602850093136, 1545271940982379, 5869487840301918, 22333881100473377, 85122472676430726, 324932398968641347, 1242132574137819311, 4754780801977591058, 18224083514440495025, 69932926170236579857, 268663970613978460710, 1033242445754706363975, 3977733225255963086695, 15328057650731416362340, 59120192070383712733335, 228224633049428050705315, 881757770639829646583480, 3409412576759142329814775, 13192826084831275350325095, 51086917518783129904345090, 197961919702850454901151945, 767610816454467010624707517, 2978352872499110051509366998, 11563138882791212911809875029, 44918930939967950176009700285, 174593200705697253200913626434, 678985059893115852427977601233, 2641918035561928489884755628085, 10284813642338883027548961771422, 40057483612905891228568746589129, 156089485797690220854261169189257, 608498303354150874059811676944094, 2373199099054809597387601987183445, 9259589649102642230389825001237581, 36143155325687815294466979403558062, 141133929022807696900357855309319686, 551318296297298226895841836739396636, 2154437013991982859151223973266017111, 8422105535902460573353212237139657836, 32935049565117341329911108368414433436, 128837480794804391294231686814232253961, 504159649159062921742208781510121479286, 1973475697954659127594313029868840013636, 7727328625870584509660917417455944315611, 30266145090034561856284314082228329357836, 118579973892046416866907966164919250795486, 464717084288790803147833983326225729898351, 1821736876318577132963206464947847525414496, 7143304966124617288790454701805892173307766, 28017294790285509721786862889706936530493411, 109916711843289013133990814886347507336349116, 431329284106342647410687786849794229058415096, 1693008447329193848007468964072511397919030261, 6646785249279335298563122340008132422018686126, 26101411097856589907818918485053183402550707016, 102521207133451069515039791076689863650678499701, 402771262148305205432086543004645400409641062766, 1582690592125375363840473527443346048315268197136, 6220483825136390437188592150230476398239269830581, 24453487413189282164379940095025144159751430013926, 96148808505504648658355837054737400502817631130056, 378121619722888386091654244926587305529231933562281, 1487310748426170392219747427190113641117644329664706, 5851304433977099638065331103013700669555026068259756, 23024100401074837095326061325803455280506150809710681, 90612879119866582000616903081680184519349032679325006, 356675117510859735951720818116813436206860188800515856, 1404202407568568839811319479559232225204936642227798531, 5529158948413130596116905382650919230499689554365543456, 21775061512814401259689768076704747282820099109326531006, 85768773466747093766508919606135428336752409847977596831, 337883579114510661146794569001349454481340388581898164796, 1331286955593172627138765270845951844271331129305687691366

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

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

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

Implicit equation for the generating function in Maple syntax:
x*(x-1)^2*(x^2-3*x+1)^2*F(x)^2-(x-1)*(2*x^2-1)*(x^2-3*x+1)^2*F(x)+x^7-2*x^6+13*x^5-12*x^4-9*x^3+16*x^2-7*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)^{2}-\left(x -1\right) \left(2 x^{2}-1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+x^{7}-2 x^{6}+13 x^{5}-12 x^{4}-9 x^{3}+16 x^{2}-7 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 67
a(6) = 224
a(7) = 751
a(n+7) = -4*(1+2*n)/(8+n)*a(n)+10*(1+3*n)/(8+n)*a(1+n)-11*(n-6)/(8+n)*a(n+2)-(296+59*n)/(8+n)*a(n+3)+(422+83*n)/(8+n)*a(n+4)-(266+45*n)/(8+n)*a(n+5)+(76+11*n)/(8+n)*a(n+6), n >= 8

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) = 67
a \! \left(6\right) = 224
a \! \left(7\right) = 751
a \! \left(n +7\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{8+n}+\frac{10 \left(1+3 n \right) a \! \left(1+n \right)}{8+n}-\frac{11 \left(n -6\right) a \! \left(n +2\right)}{8+n}-\frac{\left(296+59 n \right) a \! \left(n +3\right)}{8+n}+\frac{\left(422+83 n \right) a \! \left(n +4\right)}{8+n}-\frac{\left(266+45 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(76+11 n \right) a \! \left(n +6\right)}{8+n}, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21456/
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[7,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[7,x]*F[8,x]
F[7,x] = x
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[10,x]^3
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]^2*F[7,x]
F[13,x] = F[14,x]+F[22,x]
F[14,x] = F[15,x]*F[17,x]*F[2,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[15,x]*F[7,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[20,x]*F[7,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[20,x]*F[7,x]
F[22,x] = F[23,x]^2*F[20,x]
F[23,x] = F[11,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_{7}\! \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_{7}\! \left(x \right) F_{8}\! \left(x \right)
F_{7}\! \left(x \right) = x
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)^{3}
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)^{2} F_{7}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{22}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right) F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{15}\! \left(x \right) F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{20}\! \left(x \right) F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23} \left(x \right)^{2} F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \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_7(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_7(x)*F_8(x))
Eq(F_7(x), x)
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_10(x)**3)
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)**2*F_7(x))
Eq(F_13(x), F_14(x) + F_22(x))
Eq(F_14(x), F_15(x)*F_17(x)*F_2(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_15(x)*F_7(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_20(x)*F_7(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_20(x)*F_7(x))
Eq(F_22(x), F_20(x)*F_23(x)**2)
Eq(F_23(x), F_11(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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, 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