0132_1032_1302_3021

Counting sequence:
1, 1, 2, 6, 20, 68, 233, 806, 2817, 9941, 35385, 126921, 458362, 1665445, 6084515, 22338733, 82379941, 305024339, 1133547642, 4226673882, 15808448807, 59292986297, 222969304775, 840483393494, 3175259135943, 12020616512392, 45594268442924, 173250346902597, 659428934653097, 2513889709359991, 9597700169948315, 36693818283208474, 140471641992409316, 538421718821586576, 2066166628448030258, 7937614185266647539, 30526118126807167349, 117513314510984072625, 452809061004456932632, 1746368318772945003757, 6741101000673487317892, 26042513184545353485216, 100687598796057144759933, 389578626623438265649575, 1508433898272842668269621, 5844620604388098427998568, 22660678665004250489739083, 87915298995152555790876564, 341287244569855462312123420, 1325651726235311818946217529, 5152083905294208110630075669, 20034170771490584348707246413, 77944618264899242050019145204, 303402421680823279843177077203, 1181577893133783299851328526740, 4603718285503436506870239987938, 17945321033531191985390249455313, 69981496196756116047670160126133, 273022495784806775695001428170497, 1065591833879244758715430839081416, 4160589693339619219277693495453103, 16251184791329979441505406204662768, 63500352443188048145618173478126667, 248212289072927467233367352554601022, 970558579518426913580666479663715121, 3796357101420815271605465778238279325, 14854392374557343178214122433081536709, 58140818395631097759111423299174305845, 227636481886282890582688034593280405426, 891524516026219124680321692022155074261, 3492623718410679133338273698368928987464, 13686547880560488232539030979179185091714, 53648344734310282787147279035231356837525, 210346756093220660258353638405889436320507, 824953226703292098066196164627378695700535, 3236189127361555214694038482165267032685624, 12698332247142473858263242301663411482475813, 49838500627231900195512616864348988124718572, 195652819628441669599691749545455265041579716, 768258867003739833670496275492763941174673061, 3017358679500063338491839550146688298674801071, 11853369700153322810679812649875719535853735491, 46574691436400747552168225786230646528000674030, 183041353414864726253000982359535789433167533185, 719510958503963814611400532690060575220862278580, 2828866999310691418721177981249989472495708503316, 11124297547184556098729543805315096522077128431665, 43753793193987879709109017555996308432963747272465, 172123443898103831275169481088388430183109408414779, 677241818612674125036679016095742455423352315211386, 2665171787448165252747651966551468320647400255960745, 10490173699209090939535077522022234342863587222501396, 41296625834696632101562152780012429667487055267877168, 162599539642965699995087113699184159500020089179205841, 640317531751610964859198606651438618626055265603400359, 2521978292018421311425707095695924841131984235343076513, 9934722521200425301194603079176936599098957668664462522, 39141476600550083984199729374149944500162474677587699197, 154235517724947002805881997397700234663015975887162167986, 607849429220565477003989243081602060988992766452174457936, 2395911636135798686512669055780316250376164053374948034205

Generating function in Maple syntax:
(-2*(x-1)^3*(x-1/2)*(x^2-x+1)*(1-4*x)^(1/2)+12*x^6-37*x^5+58*x^4-51*x^3+27*x^2-8*x+1)/(8*x^6-18*x^5+20*x^4-10*x^3+2*x^2)

Generating function in latex syntax:
\frac{-2 \left(x -1\right)^{3} \left(x -\frac{1}{2}\right) \left(x^{2}-x +1\right) \sqrt{1-4 x}+12 x^{6}-37 x^{5}+58 x^{4}-51 x^{3}+27 x^{2}-8 x +1}{8 x^{6}-18 x^{5}+20 x^{4}-10 x^{3}+2 x^{2}}

Generating function in sympy syntax:
(12*x**6 - 37*x**5 + 58*x**4 - 51*x**3 + 27*x**2 - 8*x - 2*sqrt(1 - 4*x)*(x - 1)**3*(x - 1/2)*(x**2 - x + 1) + 1)/(8*x**6 - 18*x**5 + 20*x**4 - 10*x**3 + 2*x**2)

Implicit equation for the generating function in Maple syntax:
x^2*(4*x^4-9*x^3+10*x^2-5*x+1)*F(x)^2+(-12*x^6+37*x^5-58*x^4+51*x^3-27*x^2+8*x-1)*F(x)+x^7+2*x^6-13*x^5+28*x^4-31*x^3+20*x^2-7*x+1 = 0

Implicit equation for the generating function in latex syntax:
x^{2} \left(4 x^{4}-9 x^{3}+10 x^{2}-5 x +1\right) F \! \left(x \right)^{2}+\left(-12 x^{6}+37 x^{5}-58 x^{4}+51 x^{3}-27 x^{2}+8 x -1\right) F \! \left(x \right)+x^{7}+2 x^{6}-13 x^{5}+28 x^{4}-31 x^{3}+20 x^{2}-7 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 68
a(6) = 233
a(7) = 806
a(8) = 2817
a(9) = 9941
a(n+9) = 16*(-1+2*n)/(11+n)*a(n)+2*(450+197*n)/(11+n)*a(2+n)-4*(39+40*n)/(11+n)*a(n+1)-(2195+609*n)/(11+n)*a(n+3)+2*(1563+319*n)/(11+n)*a(n+4)-(2855+463*n)/(11+n)*a(n+5)+6*(281+38*n)/(11+n)*a(n+6)-6*(103+12*n)/(11+n)*a(n+7)+(127+13*n)/(11+n)*a(n+8), 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) = 20
a \! \left(5\right) = 68
a \! \left(6\right) = 233
a \! \left(7\right) = 806
a \! \left(8\right) = 2817
a \! \left(9\right) = 9941
a \! \left(n +9\right) = \frac{16 \left(-1+2 n \right) a \! \left(n \right)}{11+n}+\frac{2 \left(450+197 n \right) a \! \left(2+n \right)}{11+n}-\frac{4 \left(39+40 n \right) a \! \left(n +1\right)}{11+n}-\frac{\left(2195+609 n \right) a \! \left(n +3\right)}{11+n}+\frac{2 \left(1563+319 n \right) a \! \left(n +4\right)}{11+n}-\frac{\left(2855+463 n \right) a \! \left(n +5\right)}{11+n}+\frac{6 \left(281+38 n \right) a \! \left(n +6\right)}{11+n}-\frac{6 \left(103+12 n \right) a \! \left(n +7\right)}{11+n}+\frac{\left(127+13 n \right) a \! \left(n +8\right)}{11+n}, \quad n \geq 10

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21275/
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[4,x]*F[9,x]
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[17,x]
F[16,x] = F[9,x]^3
F[17,x] = F[18,x]^2*F[2,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[12,x]*F[18,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_{4}\! \left(x \right) F_{9}\! \left(x \right)
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_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{9} \left(x \right)^{3}
F_{17}\! \left(x \right) = F_{18} \left(x \right)^{2} F_{2}\! \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)
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_4(x)*F_9(x))
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_17(x))
Eq(F_16(x), F_9(x)**3)
Eq(F_17(x), F_18(x)**2*F_2(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_12(x)*F_18(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, 0, 2], "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, 0, 2], "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, 0, 2], "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"}}, {"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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"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": [0, 1, 3, 2], "pos": [[0, 0], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"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": [0, 1, 3, 2], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 2], [0, 0], [0, 2], [0, 2]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"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]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "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, 1, 3, 2], 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