0132_0213_0231_0312_1302

Counting sequence:
1, 1, 2, 6, 19, 61, 199, 660, 2223, 7593, 26262, 91850, 324427, 1155980, 4150950, 15008329, 54598228, 199711783, 734113657, 2710475604, 10047689248, 37382208143, 139540886770, 522459503444, 1961597543268, 7383755780851, 27859264660781, 105344270066733, 399147279271753, 1515218996772818, 5762124857702841, 21948522371965753, 83733310503324902, 319904732070937287, 1223869713765792886, 4688217280518297349, 17980718666400983947, 69040579382267748378, 265383213416445663704, 1021150434842508769561, 3933061326386027477270, 15162664983081377434190, 58506592843547617133810, 225943837654892749611249, 873264559280924003218655, 3377731713463466281590913, 13074461785624956622666751, 50644017562144456430206552, 196302266883212119028841849, 761383172921769867706328702, 2954953958740262090421508863, 11475113929201276260194852935, 44587396463025081330627474754, 173343112253305666134448964740, 674266386913050754403227509697, 2624088276346079396557387109298, 10217376647613989033084962668904, 39802178255083752670557944960612, 155122069081007744082217112110363, 604829342077750857905181420186725, 2359272873931287274640517179029785, 9206687886363392533945655347408117, 35942042481028310625878360039307066, 140368808918203321750634854152493574, 548405383853099569762140128623574363, 2143339600913042801019897325736774921, 8379799567150375820702157228814605131, 32773666964659963201436854374973232565, 128221485993529821285860492166904150598, 501807021373094655798442519883097188323, 1964485355539030947657392023811717750360, 7692954039237992597632144245468493059493, 30134644179582521235398875715529616531179, 118076656197421436176375410164189486267318, 462789691301908548410067683497467885759810, 1814352636841196961742097241748436168778889, 7115001359867970158918655808396029156581698, 27908759334506844204886943904328618224501880, 109500332696902021143472910674226681169087562, 429731244113543916421124800717889587858221663, 1686872780820982016880995367606318877100056581, 6623218179501894968493731847006678323758846899, 26010855837748538817593494607235650923464481497, 102173125796257935728520905626359569549356951754, 401432813702869548364789978439393024440716555769, 1577542201713137300026099677792338608661378088602, 6200673781088857908658654097718258636408845008617, 24377237573726318203633372086102098311393469204971, 95855227822109690583484699849356605139810105198358, 376990921430543425181248263252776472180678454626548, 1482954701933491330236168936646100065437020560716663, 5834517912668885405440194504058710796819083227990992, 22959393981326203377509131424136473964739170480970690, 90363391703275527525967253660004247022769830985478618, 355712927636516840729159275960041331096698607568482875, 1400490645505854398662653609650284127171400219217981649, 5514836966284858922701454539302183090817800733068339127, 21719786823576996183283307149656594175130346808469334275, 85555397113150981834960684783907680718740214238684656080, 337059707084432407947705517383888197537862616900979971299, 1328105224393939046946070538545749168292443496450796870588

Generating function in Maple syntax:
(x^2+2*x-1+(x-1)^2*(-4*x+1)^(1/2))*(x-1/2)/x/(x^4-4*x^3+8*x^2-5*x+1)

Generating function in latex syntax:
\frac{\left(x^{2}+2 x -1+\left(x -1\right)^{2} \sqrt{-4 x +1}\right) \left(x -\frac{1}{2}\right)}{x \left(x^{4}-4 x^{3}+8 x^{2}-5 x +1\right)}

Generating function in sympy syntax:
(x - 1/2)*(x**2 + 2*x + sqrt(1 - 4*x)*(x - 1)**2 - 1)/(x*(x**4 - 4*x**3 + 8*x**2 - 5*x + 1))

Implicit equation for the generating function in Maple syntax:
x*(x^4-4*x^3+8*x^2-5*x+1)*F(x)^2-(2*x-1)*(x^2+2*x-1)*F(x)+(2*x-1)^2 = 0

Implicit equation for the generating function in latex syntax:
x \left(x^{4}-4 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)^{2}-\left(2 x -1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+\left(2 x -1\right)^{2} = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 61
a(6) = 199
a(n+7) = 4*(2*n+3)/(n+8)*a(n)-2*(23*n+52)/(n+8)*a(1+n)+(363+127*n)/(n+8)*a(n+2)-(181*n+688)/(n+8)*a(n+3)+2*(69*n+334)/(n+8)*a(n+4)-(57*n+335)/(n+8)*a(n+5)+(83+12*n)/(n+8)*a(n+6), 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) = 61
a \! \left(6\right) = 199
a \! \left(n +7\right) = \frac{4 \left(2 n +3\right) a \! \left(n \right)}{n +8}-\frac{2 \left(23 n +52\right) a \! \left(1+n \right)}{n +8}+\frac{\left(363+127 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(181 n +688\right) a \! \left(n +3\right)}{n +8}+\frac{2 \left(69 n +334\right) a \! \left(n +4\right)}{n +8}-\frac{\left(57 n +335\right) a \! \left(n +5\right)}{n +8}+\frac{\left(83+12 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21545/
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[0,x]*F[12,x]*F[7,x]
F[7,x] = F[13,x]+F[8,x]
F[8,x] = F[9,x]^2
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[9,x]
F[15,x] = F[16,x]+F[19,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[16,x]
F[19,x] = F[17,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[24,x]
F[21,x] = 0
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[17,x]+F[20,x]
F[24,x] = F[12,x]*F[19,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_{0}\! \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)^{2}
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_{9}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \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_0(x)*F_12(x)*F_7(x))
Eq(F_7(x), F_13(x) + F_8(x))
Eq(F_8(x), F_9(x)**2)
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)*F_9(x))
Eq(F_15(x), F_16(x) + F_19(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_16(x))
Eq(F_19(x), F_17(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_24(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_17(x) + F_20(x))
Eq(F_24(x), F_12(x)*F_19(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": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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]]}], "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": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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]]}], "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": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "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, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "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, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "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}}, {"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": [0, 2, 1, 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