0132_0231_1032_1320_2031_3021

Counting sequence:
1, 1, 2, 6, 18, 53, 159, 494, 1591, 5283, 17970, 62271, 218926, 778551, 2794638, 10109650, 36814078, 134825433, 496254807, 1834705199, 6810098657, 25368551257, 94808679211, 355374205263, 1335674530575, 5032657237133, 19006068168217, 71930182808385, 272764463617725, 1036245995120301, 3943511165187673, 15031429672916802, 57381369966419439, 219358933996258915, 839686981341359866, 3218282636731075507, 12349385865023583286, 47440926191863893063, 182440804554415043450, 702308931817770525651, 2706138025078014436392, 10436790860077593718775, 40286588837987166508512, 155637051831637349097197, 601739024512755375173558, 2328256521204316907203051, 9015020620077035774031268, 34930319026753495606560303, 135433414726845240793566714, 525442061590974797727762539, 2039804547851461513285114730, 7923309532535870547846689965, 30794224024900332028744144084, 119747723409773015848718620983, 465899150527322370850474991838, 1813572796599340965950002168457, 7062991726782335725912018783244, 27519823992687216744659283508459, 107275082378618408371355073042694, 418351825098501739092289566660825, 1632178955305248369034047530236308, 6370465570321309997968670620985399, 24874029970936984468288382937692882, 97159948367414361344755169037912586, 379655201223794921753480271959255614, 1484046275781279440996167036403717481, 5803056809031826001366420164296908727, 22699326938145000892682265468360947831, 88820171180016659005779795881814757601, 347655076540700551458535794890250173353, 1361193734322151566550997553751394149219, 5331162526418319861513058538935580898363, 20885789469862175146908111580208918235663, 81847117032030996378152583298866243318725, 320830804489503888488045401654601282705501, 1257956769509935727858124778422718917345865, 4933660723709094553882535815203804799554145, 19354582076733012265437277460775217508209077, 75946266172463302009435484799153174242369853, 298081095877566762774914760877072606578235787, 1170213727517819039598436292898138749808935977, 4595112578038516612936058482027593088354629129, 18047826003321951227468557548549498577522929795, 70900460158087683634814306251329347891638589769, 278590538152293241972226004915181872789525713179, 1094900443138379371768015045448397255606904532961, 4303994908008617876868961949092682851487183169267, 16922155214392902664011350949897241616054721906241, 66546380213688046797999839344349707307787265131991, 261743637253680110246532373019911478430634042169233, 1029695753906453703744123135492493342685163471955279, 4051548321432286129695219202662833727210565671541653, 15944493003386354170392288088885211656995912825674483, 62759045410081418615642520070650463031549756851110825, 247067860320248550819429583288714871655613208256884391, 972812016529522185696581856735253501799089972754624855, 3831006003243311211352771275807226127017383574153416015, 15089210080770272115919619115759674828656172516610701117, 59441330518022822522056888302082141226002605036024672953, 234194938170837536848219045095658109589903887352516894697, 922852118751055981757116769705885054072557350173873247517

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

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

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

Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^4*F(x)^2+(2*x-1)*(2*x^5-2*x^4-x^3+5*x^2-4*x+1)*(x-1)^2*F(x)+x^9+2*x^8+3*x^7-14*x^6+2*x^5+30*x^4-42*x^3+26*x^2-8*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(2 x -1\right) \left(2 x^{5}-2 x^{4}-x^{3}+5 x^{2}-4 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{9}+2 x^{8}+3 x^{7}-14 x^{6}+2 x^{5}+30 x^{4}-42 x^{3}+26 x^{2}-8 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 53
a(6) = 159
a(7) = 494
a(8) = 1591
a(9) = 5283
a(n+7) = -8*(-1+2*n)/(8+n)*a(n)+4*(-9+5*n)/(8+n)*a(1+n)+2*(83+20*n)/(8+n)*a(n+2)-(405+107*n)/(8+n)*a(n+3)+5*(93+20*n)/(8+n)*a(n+4)-(268+47*n)/(8+n)*a(n+5)+(75+11*n)/(8+n)*a(n+6)-1/(8+n), 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) = 53
a \! \left(6\right) = 159
a \! \left(7\right) = 494
a \! \left(8\right) = 1591
a \! \left(9\right) = 5283
a \! \left(n +7\right) = -\frac{8 \left(-1+2 n \right) a \! \left(n \right)}{8+n}+\frac{4 \left(-9+5 n \right) a \! \left(1+n \right)}{8+n}+\frac{2 \left(83+20 n \right) a \! \left(n +2\right)}{8+n}-\frac{\left(405+107 n \right) a \! \left(n +3\right)}{8+n}+\frac{5 \left(93+20 n \right) a \! \left(n +4\right)}{8+n}-\frac{\left(268+47 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(75+11 n \right) a \! \left(n +6\right)}{8+n}-\frac{1}{8+n}, \quad n \geq 10

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22371/
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[9,x]^3
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[22,x]
F[16,x] = F[17,x]*F[19,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[12,x]*F[17,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[17,x]*F[19,x]
F[22,x] = F[10,x]*F[23,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[12,x]*F[23,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_{9} \left(x \right)^{3}
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_{22}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \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_9(x)**3)
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_22(x))
Eq(F_16(x), F_17(x)*F_19(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_12(x)*F_17(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_17(x)*F_19(x))
Eq(F_22(x), F_10(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_12(x)*F_23(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, 3, 1], "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, 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, 3, 1], "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, 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, 3, 1], "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, 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"}}, {"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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 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, 1, 3, 2], "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, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [1, 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": []}, {"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], 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