0132_0213_0231_0312_1320

Counting sequence:
1, 1, 2, 6, 19, 60, 190, 611, 2007, 6734, 23019, 79917, 281024, 998742, 3581273, 12939832, 47063087, 172160908, 632992510, 2337921848, 8670119662, 32270883442, 120514393658, 451420234750, 1695608007382, 6385197242258, 24101344485936, 91169547338438, 345566590775148, 1312278900761718, 4992041845763576, 19021228765158099, 72587611155743411, 277401906965291770, 1061554819134636143, 4067501316863533949, 15603943122557517288, 59928583785154370722, 230409589884989483945, 886767717240699313761, 3416171879256829141810, 13172537367054691246308, 50836999161665854018225, 196360132399466281725851, 759054726995919166429066, 2936457768986354628949210, 11368178269452801523848023, 44041379442093864685065775, 170734465457335563344408506, 662307753290978576697386228, 2570784402899989249527798403, 9984541690255679709366458485, 38800504057398725184579985490, 150863441796655172848254105070, 586893425378200414148959308901, 2284308350521749938649152890487, 8895340359276114178715052682324, 34655749735354476023584837350844, 135078375186127938839796214001291, 526729247427854681912652483893581, 2054819742443980458682825543606066, 8019340317637712894460100417800866, 31309519323978576167140205969607877, 122287339732808079555565284946610176, 477802803213557369871648079932204575, 1867553228436064821856507694534766124, 7302130867592074145297174742408576286, 28561039232586155332213462169106555536, 111748528828042501529212366050233961574, 437369880346786535376815655736750772050, 1712344690669977815906479663645090986914, 6706021714578081701272458598766907608866, 26270389076019974576510036381068184996506, 102941936183404865392882496244504539992034, 403495850283199354013276532847981536458140, 1581989130428057704020748340672611894492114, 6204152975826048936746858398426128979164236, 24337339398128682390317596205330427194872726, 95493196640228680488349136932362822652798924, 374781149790985558687901420802896055390946400, 1471249633697272139826345330391483818666870234, 5776912130560523853307515503479778825169754046, 22688378555935138808843231249204995316850363726, 89126548664034501961191878351388980766639886404, 350190501800710967098677907045376739403070540600, 1376236503806310144158795229052548430833592218054, 5409673960775532902999095372042948881833308808066, 21268475472246161731037370978270486371031336712156, 83634775612377683794522437123196830884122669334564, 328943098031392367243985510439840512498854758718330, 1294005591352083156617611698961177852376267970631554, 5091329754480368489788876841875150289810148820584496, 20035688157066367878251764442283780797063325657183956, 78859421333563781808476428464290815748078501030022534, 310439923831320443220777679837467187030156678562574430, 1222291063825502213165896963088370725654234723216767494, 4813303721550736419349752442465079501099253071495966326, 18957540004343580327416765648155733475786026033099788226, 74677427317960448463909783329630465527814786778201816544, 294214436132668409875111950420175217327477043992731055470, 1159323196814123033850041019201104765008956312167671890644

Generating function in Maple syntax:
1/2*((-x^4+3*x^3-2*x+1)*(1-4*x)^(1/2)+5*x^4-7*x^3+2*x^2+2*x-1)/x/(x-1)^3

Generating function in latex syntax:
\frac{\left(-x^{4}+3 x^{3}-2 x +1\right) \sqrt{1-4 x}+5 x^{4}-7 x^{3}+2 x^{2}+2 x -1}{2 x \left(x -1\right)^{3}}

Generating function in sympy syntax:
(5*x**4/2 - 7*x**3/2 + x**2 + x + sqrt(1 - 4*x)*(-x**4 + 3*x**3 - 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^4-7*x^3+2*x^2+2*x-1)*(x-1)^3*F(x)+x^8-7*x^6+19*x^5-17*x^4+x^3+8*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}-7 x^{3}+2 x^{2}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{8}-7 x^{6}+19 x^{5}-17 x^{4}+x^{3}+8 x^{2}-5 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 60
a(6) = 190
a(7) = 611
a(8) = 2007
a(n+6) = 2*(-1+2*n)/(n+7)*a(n)-(9+17*n)/(n+7)*a(n+1)+2*(1+8*n)/(n+7)*a(n+2)+(41+5*n)/(n+7)*a(n+3)-2*(36+7*n)/(n+7)*a(n+4)+(41+7*n)/(n+7)*a(n+5)+2*(n-3)/(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) = 19
a \! \left(5\right) = 60
a \! \left(6\right) = 190
a \! \left(7\right) = 611
a \! \left(8\right) = 2007
a \! \left(n +6\right) = \frac{2 \left(-1+2 n \right) a \! \left(n \right)}{n +7}-\frac{\left(9+17 n \right) a \! \left(n +1\right)}{n +7}+\frac{2 \left(1+8 n \right) a \! \left(n +2\right)}{n +7}+\frac{\left(41+5 n \right) a \! \left(n +3\right)}{n +7}-\frac{2 \left(36+7 n \right) a \! \left(n +4\right)}{n +7}+\frac{\left(41+7 n \right) a \! \left(n +5\right)}{n +7}+\frac{2 n -6}{n +7}, \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21546/
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[9,x]
F[4,x] = F[10,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[5,x]^2*F[9,x]
F[9,x] = x
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[5,x]*F[9,x]
F[12,x] = F[13,x]+F[4,x]
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[18,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[9,x]
F[17,x] = F[1,x]+F[15,x]
F[18,x] = F[19,x]+F[20,x]
F[19,x] = F[17,x]^2
F[20,x] = F[21,x]
F[21,x] = F[17,x]*F[5,x]*F[7,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_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{4}\! \left(x \right) = F_{10}\! \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_{5} \left(x \right)^{2} F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{5}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17} \left(x \right)^{2}
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right) F_{5}\! \left(x \right) F_{7}\! \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_4(x)*F_9(x))
Eq(F_4(x), F_10(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_5(x)**2*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_5(x)*F_9(x))
Eq(F_12(x), F_13(x) + F_4(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_18(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_9(x))
Eq(F_17(x), F_1(x) + F_15(x))
Eq(F_18(x), F_19(x) + F_20(x))
Eq(F_19(x), F_17(x)**2)
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_17(x)*F_5(x)*F_7(x)*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": [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, 2, 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, 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, 2, 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, 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, 2, 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"}}, {"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], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [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, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 2], [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, 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], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}], "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], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 1]]}], "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], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": 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