0132_1032_1320_3021
Counting sequence:
1, 1, 2, 6, 20, 66, 214, 694, 2280, 7625, 25957, 89763, 314573, 1114779, 3987809, 14379766, 52210858, 190711360, 700303312, 2583637789, 9571879441, 35595950248, 132826887910, 497186423467, 1866310454203, 7023895560832, 26497942823586, 100185992096615, 379570235026003, 1440803295421254, 5478834942582152, 20868532780787082, 79610412592275104, 304144841861953081, 1163551952368475269, 4457088778846793999, 17094064955439929025, 65635525483843235407, 252293149835345255069, 970778699767343733687, 3739044147146768911937, 14414696292037902167025, 55620571786018828198351, 214799037314826917299405, 830193394114191907355287, 3211149677512624826888797, 12429723538980065871889015, 48146895926699371256461891, 186624281868552485529206065, 723850643385997961382014719, 2809308264920980462190249725, 10909598769382259888061341209, 42390356011387054982931074071, 164802874296019568651492215663, 641051605368713501907226859017, 2494843149459194152613003726005, 9714211495188442990332096312667, 37842371824850508303847553651983, 147485205157052548086254261437537, 575057238136982720029551083377597, 2243157354180434229703663013949931, 8753632545529069384115520202371691, 34173617132620955418116980847444557, 133463371080786337218350495764929046, 521430505843917132771953185237278946, 2037928452907578069782268079272976120, 7967732783780557537727399449293468856, 31162286413642528538020641092381996029, 121918102858527077283390414895749095089, 477141506248655831142723127118562548536, 1867937292442192117410302198226900514870, 7314920794730856984070979990003773512991, 28654016742973593111093198806150336821399, 112275862361805432738669239897427043454044, 440056965362756270033195856296999032582966, 1725240923773848346556008481550073652164255, 6765592325749176626934995648250844132410931, 26538361959059525939987371286110357874484994, 104124211342530543808828936052492833319801284, 408635299361468326354402605241463691238588891, 1604072161747468270298558052003136408934488455, 6298153118265698246039723608294007137121740226, 24734397581242627735767067614226466807154331804, 97159635776932625904770584274123828618795491119, 381737242016816502893646623043740619253474016231, 1500151338258403286513042007161209989825228770700, 5896514340503008769941262675551076686386849718166, 23181595100171294158540495465221556879869885384229, 91154254562056997517101214621761751555674977082929, 358504251137491289058709085910782169476488415937926, 1410241679200364600894976709890642427122664512583380, 5548464935374833253052836729041979429411855619295151, 21833857031807370742431718456224516365882803802604239, 85933936135735327458990230177768807255889782202176244, 338278142383414421064753656214260777708973693889297166, 1331853862662859038954797677709309564396027201605497255, 5244584446191433098252493671953640310910301840434932711, 20655512452765045067285953262966855853503561850823895284, 81363534694568635650309287871143396133148316342089758574, 320546607793592894945058649735046988922139522253682587571, 1263044790828965174585722613897378233926968470750443623815
Generating function in Maple syntax:
1/2*((-x^4+2*x^3+x^2-2*x+1)*(1-4*x)^(1/2)+5*x^4-4*x^3+x^2+2*x-1)/x/(x-1)^3
Generating function in latex syntax:
\frac{\left(-x^{4}+2 x^{3}+x^{2}-2 x +1\right) \sqrt{1-4 x}+5 x^{4}-4 x^{3}+x^{2}+2 x -1}{2 x \left(x -1\right)^{3}}
Generating function in sympy syntax:
(5*x**4/2 - 2*x**3 + x**2/2 + x + sqrt(1 - 4*x)*(-x**4 + 2*x**3 + x**2 - 2*x + 1)/2 - 1/2)/(x*(x - 1)**3)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^6*F(x)^2-(5*x^4-4*x^3+x^2+2*x-1)*(x-1)^3*F(x)+x^8+2*x^7-7*x^6+14*x^5-8*x^4-4*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{6} F \! \left(x \right)^{2}-\left(5 x^{4}-4 x^{3}+x^{2}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{8}+2 x^{7}-7 x^{6}+14 x^{5}-8 x^{4}-4 x^{3}+9 x^{2}-5 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 214
a(7) = 694
a(8) = 2280
a(n+6) = 2*(-1+2*n)/(n+7)*a(n)-(3+13*n)/(n+7)*a(n+1)+(-13+7*n)/(n+7)*a(n+2)+(53+11*n)/(n+7)*a(n+3)-15*(n+5)/(n+7)*a(n+4)+(41+7*n)/(n+7)*a(n+5)+6*n/(n+7), n >= 9
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) = 66
a \! \left(6\right) = 214
a \! \left(7\right) = 694
a \! \left(8\right) = 2280
a \! \left(n +6\right) = \frac{2 \left(-1+2 n \right) a \! \left(n \right)}{n +7}-\frac{\left(3+13 n \right) a \! \left(n +1\right)}{n +7}+\frac{\left(-13+7 n \right) a \! \left(n +2\right)}{n +7}+\frac{\left(53+11 n \right) a \! \left(n +3\right)}{n +7}-\frac{15 \left(n +5\right) a \! \left(n +4\right)}{n +7}+\frac{\left(41+7 n \right) a \! \left(n +5\right)}{n +7}+\frac{6 n}{n +7}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21278/
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[12,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[12,x]*F[7,x]
F[7,x] = F[13,x]+F[8,x]
F[8,x] = F[9,x]^3
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[9,x]^2*F[12,x]
F[12,x] = x
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]
F[15,x] = F[16,x]+F[20,x]
F[16,x] = F[17,x]+F[8,x]
F[17,x] = F[18,x]^2*F[10,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[12,x]*F[18,x]
F[20,x] = F[18,x]^2*F[10,x]*F[9,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_{12}\! \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_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9} \left(x \right)^{3}
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18} \left(x \right)^{2} F_{10}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18} \left(x \right)^{2} F_{10}\! \left(x \right) F_{9}\! \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_12(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_12(x)*F_7(x))
Eq(F_7(x), F_13(x) + F_8(x))
Eq(F_8(x), F_9(x)**3)
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x)**2)
Eq(F_12(x), x)
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x))
Eq(F_15(x), F_16(x) + F_20(x))
Eq(F_16(x), F_17(x) + F_8(x))
Eq(F_17(x), F_10(x)*F_18(x)**2)
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_12(x)*F_18(x))
Eq(F_20(x), F_10(x)*F_18(x)**2*F_9(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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": [3, 0, 2, 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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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": [3, 0, 2, 1], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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": [3, 0, 2, 1], "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 0], [0, 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": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 2], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 2], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": 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