0132_0213_0231_1320_2031_3021

Counting sequence:
1, 1, 2, 6, 18, 52, 154, 478, 1549, 5184, 17751, 61805, 217958, 776570, 2790621, 10101550, 36797800, 134792786, 496189408, 1834574281, 6809836685, 25368027160, 94807630846, 355372108343, 1335670336525, 5032648848802, 19006051391302, 71930149254279, 272764396509213, 1036245860902952, 3943510896752624, 15031429136046326, 57381368892678081, 219358931848775764, 839686977046393099, 3218282628141141477, 12349385847843714698, 47440926157504155326, 182440804485695567381, 702308931680331572883, 2706138024803136530190, 10436790859527837905668, 40286588836887654881557, 155637051829438325842507, 601739024508357328663358, 2328256521195520814181790, 9015020620059443587987843, 34930319026718311234472507, 135433414726774872049390132, 525442061590834060239408340, 2039804547851180038308405251, 7923309532535307597893269879, 30794224024899206128837302736, 119747723409770764048904937062, 465899150527317867250847622721, 1813572796599331958750747428897, 7062991726782317711513509302746, 27519823992687180715862264546032, 107275082378618336313761035116355, 418351825098501594977101490806607, 1632178955305248080803671378526276, 6370465570321309421507918317563682, 24874029970936983315366878330847737, 97159948367414359038912159824220526, 379655201223794917141794253531869664, 1484046275781279431772794999548943690, 5803056809031825982919676090587359192, 22699326938145000855788777320941846745, 88820171180016658931992819586976553349, 347655076540700551310961842300573762704, 1361193734322151566255849648572041325710, 5331162526418319860922762728576875249067, 20885789469862175145727519959491506934725, 81847117032030996375791400057431420714434, 320830804489503888483323035171731637494434, 1257956769509935727848680045456979626921175, 4933660723709094553863646349272326218702137, 19354582076733012265399498528912260346502360, 75946266172463302009359926935427259918953644, 298081095877566762774763645149620777931400519, 1170213727517819039598134061443235092515262515, 4595112578038516612935454019117785773767279202, 18047826003321951227467348622729883948348226860, 70900460158087683634811888399690118633289180739, 278590538152293241972221169211903414272826891879, 1094900443138379371768005374041840338573506887040, 4303994908008617876868942606279569017420387874022, 16922155214392902664011312264271013947921131312265, 66546380213688046797999761973097251971520083940469, 261743637253680110246532218277406567758099679782534, 1029695753906453703744122826007483521340094747178140, 4051548321432286129695218583692814084520428221983547, 15944493003386354170392286850945172371615637926554355, 62759045410081418615642517594770384460789207052866564, 247067860320248550819429578336954714514092108660391774, 972812016529522185696581846831733187516047773561635435, 3831006003243311211352771256000185498451299175767432897, 15089210080770272115919619076145593571524003719838730510, 59441330518022822522056888222853978711738267442480727274, 234194938170837536848219044937201784561375212165428998779, 922852118751055981757116769388972404015499999799697451025

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

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

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

Implicit equation for the generating function in Maple syntax:
x*(-1+x)^6*F(x)^2+(2*x^5-x^4+3*x^2-3*x+1)*(-1+x)^3*F(x)+x^9-3*x^6+3*x^5+8*x^4-18*x^3+15*x^2-6*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(-1+x \right)^{6} F \! \left(x \right)^{2}+\left(2 x^{5}-x^{4}+3 x^{2}-3 x +1\right) \left(-1+x \right)^{3} F \! \left(x \right)+x^{9}-3 x^{6}+3 x^{5}+8 x^{4}-18 x^{3}+15 x^{2}-6 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 52
a(6) = 154
a(7) = 478
a(8) = 1549
a(9) = 5184
a(n+5) = -2*(2*n-1)/(n+6)*a(n)+(-11+n)/(n+6)*a(n+1)+3*(4*n+11)/(n+6)*a(n+2)-3*(5*n+18)/(n+6)*a(n+3)+(33+7*n)/(n+6)*a(n+4)-3/2*(n^2-3*n+6)/(n+6), n >= 10

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) = 18
a \! \left(5\right) = 52
a \! \left(6\right) = 154
a \! \left(7\right) = 478
a \! \left(8\right) = 1549
a \! \left(9\right) = 5184
a \! \left(n +5\right) = -\frac{2 \left(2 n -1\right) a \! \left(n \right)}{n +6}+\frac{\left(-11+n \right) a \! \left(n +1\right)}{n +6}+\frac{3 \left(4 n +11\right) a \! \left(n +2\right)}{n +6}-\frac{3 \left(5 n +18\right) a \! \left(n +3\right)}{n +6}+\frac{\left(33+7 n \right) a \! \left(n +4\right)}{n +6}-\frac{3 \left(n^{2}-3 n +6\right)}{2 \left(n +6\right)}, \quad n \geq 10

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22213/
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[8,x]
F[4,x] = F[5,x]+F[9,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[5,x]^2*F[8,x]
F[8,x] = x
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[8,x]
F[11,x] = F[12,x]+F[21,x]
F[12,x] = F[13,x]*F[5,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[5,x]^2
F[15,x] = F[16,x]*F[18,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[16,x]*F[8,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[8,x]
F[20,x] = F[1,x]+F[19,x]
F[21,x] = F[16,x]*F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[16,x]*F[18,x]*F[8,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \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_{5} \left(x \right)^{2} F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{21}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{5}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{5} \left(x \right)^{2}
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right) F_{18}\! \left(x \right) F_{8}\! \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_8(x))
Eq(F_4(x), F_5(x) + F_9(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_5(x)**2*F_8(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_8(x))
Eq(F_11(x), F_12(x) + F_21(x))
Eq(F_12(x), F_13(x)*F_5(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_5(x)**2)
Eq(F_15(x), F_16(x)*F_18(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_16(x)*F_8(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_8(x))
Eq(F_20(x), F_1(x) + F_19(x))
Eq(F_21(x), F_16(x)*F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_16(x)*F_18(x)*F_8(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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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": [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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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": [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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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], [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": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 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, 1], "pos": [[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], [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": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 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, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 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