0132_0213_0231_0312_1320_3021

Counting sequence:
1, 1, 2, 6, 18, 52, 157, 497, 1631, 5493, 18858, 65711, 231734, 825409, 2964962, 10728268, 39065534, 143047500, 526399081, 1945668362, 7220164293, 26889574084, 100469991125, 376513308156, 1414840405471, 5329942273250, 20125253091275, 76153397336580, 288735151243089, 1096762967060102, 4173255204251399, 15905146958037373, 60709562297950967, 232056263872372789, 888198240486908642, 3403875823656722251, 13060332756759605246, 50167624190428195381, 192910481038673607910, 742552794306264449265, 2860988327639787459096, 11033205248731169284687, 42585843611274207218728, 164508811496093013054209, 635999881795908798440790, 2460671095485245530267171, 9527182072534167540147812, 36912744972269289109784703, 143112198168780781091393434, 555205318863419600554439597, 2155242629682132481601597382, 8371318887619905351263247655, 32533961820634375569798623468, 126507452001158500255440867993, 492178465519102163402638949050, 1915791235555749997198063161479, 7460793264819758095202851259244, 29068703810358555372890893583569, 113308668170550880999693649947298, 441866155865780414601083995414995, 1723860585623445775590711649898860, 6728082574739758369121663161443413, 26269543323796649587032813639539286, 102607784098107858199743820886862436, 400930841158198164716528937200302574, 1567166094992073153897945173677001100, 6127908276277282879498380027995827801, 23969370286824407276735506830447498002, 93787250848388488274556510585940801653, 367087554381600695568513215991355267804, 1437243084290627276227857240390227296205, 5628876403026666540713639050650122092656, 22051618976510243114713639602099926946607, 86413787876524326652411818860189620521758, 338724067212259669811668871314904126690199, 1328086332696612538344101539436406392338480, 5208596486940640958981263317342117856599221, 20432725937817600434692999110042760719615762, 80175235752903478230500175893997409226725843, 314673216043445803800305231347928392445032514, 1235327956427049634737344414177752411087589825, 4850708509643298744723236680708708905184022856, 19051363053219091368926768778122898814748216337, 74841509220370829724542921592382023504836842898, 294071104142602529880032473364797598418919151519, 1155721797158261442532314720153774137864336481700, 4543005618831907459102616353570373703990655671941, 17861594082863083357630102144205828847652598157082, 70239619907517095665572290497760031625395294337463, 276265832460420404616098238751218383392628721498004, 1086809535542537304392040685995064782131223974683205, 4276211326630188004404448449000714074629341936782806, 16828391516338427508044321090552249661231702474545407, 66237222462770958537796411280366904830961531934171608, 260757049267999437362377951286792322177286561642798809, 1026698378262691055797530608699084844900206221296911260, 4043161130209294238677711056246868630778762987928931111, 15924621079910343497983324231979594329619441999684895962, 62731492997788080228718017534744664561636725370920446163, 247154895078545647545366735169047694945370631298604637604, 973909369444049152352904965455502329744187201860566110585

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

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

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

Implicit equation for the generating function in Maple syntax:
x*(x-1)^4*F(x)^2-(2*x^5+2*x^4-3*x^3+2*x^2-2*x+1)*(x-1)^2*F(x)+x^9+2*x^8-2*x^7-2*x^5+5*x^4-7*x^3+7*x^2-4*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{4} F \! \left(x \right)^{2}-\left(2 x^{5}+2 x^{4}-3 x^{3}+2 x^{2}-2 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{9}+2 x^{8}-2 x^{7}-2 x^{5}+5 x^{4}-7 x^{3}+7 x^{2}-4 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) = 157
a(7) = 497
a(8) = 1631
a(n+4) = -2*(-1+2*n)/(5+n)*a(n)+(5+9*n)/(5+n)*a(1+n)-2*(12+5*n)/(5+n)*a(n+2)+2*(11+3*n)/(5+n)*a(n+3)-3*(1+n)/(5+n), 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) = 18
a \! \left(5\right) = 52
a \! \left(6\right) = 157
a \! \left(7\right) = 497
a \! \left(8\right) = 1631
a \! \left(n +4\right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{5+n}+\frac{\left(5+9 n \right) a \! \left(1+n \right)}{5+n}-\frac{2 \left(12+5 n \right) a \! \left(n +2\right)}{5+n}+\frac{2 \left(11+3 n \right) a \! \left(n +3\right)}{5+n}-\frac{3 \left(1+n \right)}{5+n}, \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22143/
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[15,x]
F[12,x] = F[5,x]^2*F[13,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[13,x]*F[8,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[5,x]^2*F[6,x]
F[17,x] = F[18,x]^2
F[18,x] = F[14,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_{15}\! \left(x \right)
F_{12}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{6}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18} \left(x \right)^{2}
F_{18}\! \left(x \right) = F_{14}\! \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_15(x))
Eq(F_12(x), F_13(x)*F_5(x)**2)
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_13(x)*F_8(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_5(x)**2*F_6(x))
Eq(F_17(x), F_18(x)**2)
Eq(F_18(x), F_14(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]]}, {"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": [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]]}, {"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": [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]]}, {"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, 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": [2, 0, 1], "pos": [[0, 1], [0, 0], [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": [2, 0, 1], "pos": [[0, 2], [0, 0], [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}}, {"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": [2, 0, 1], "pos": [[0, 1], [0, 0], [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": [2, 0, 1], "pos": [[0, 1], [0, 0], [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"}}, {"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"}}, {"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": 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