0213_1032_1302_2130

Counting sequence:
1, 1, 2, 6, 20, 66, 214, 691, 2247, 7403, 24766, 84134, 289953, 1012325, 3575248, 12754719, 45904401, 166480034, 607807615, 2232029045, 8238576985, 30546418127, 113710921056, 424805735025, 1592084223442, 5984033933386, 22550637292362, 85184564127602, 322488461899471, 1223329840044956, 4649253169623489, 17700066240418736, 67494282218366251, 257758684258603398, 985762749788010111, 3774912259491360846, 14473806880890244568, 55560901078051097254, 213519821105243016784, 821417012557188269367, 3163166309276544130680, 12192459141014173111125, 47038267067419769207427, 181628209788909739641499, 701891331476506001561405, 2714531333433676326780204, 10506139390193672794314269, 40691218259367094721021030, 157708204512042918619503944, 611633976759647289959586584, 2373564628057329999490390836, 9216622204922392024295667413, 35809111374430852962431678086, 139205596798736775854892709827, 541442264619942565400128169673, 2107032969998249849693012728111, 8203630979458617723388080319841, 31955736475898028260088582667620, 124535222886810633561122597055443, 485544830068797617581655420953826, 1893885337140291303499224704931566, 7390248245694597748687169491190786, 28849577584690348131165207860999082, 112665085527289900766928205309228556, 440152674874128948685268438155803166, 1720189672388132337637425434335846884, 6725172844519095348661166868091306128, 26301471051275059056542100070197106918, 102896750495328957234632722868843670971, 402683713810903592746492866305127664668, 1576388265794452160522629905353480077085, 6172984062076274292118170627707995122119, 24179996151273804139980121783838598727235, 94742063356833752216282795648146412395631, 371322757247563821314054678493789096059976, 1455724280434680748478662818896531638800659, 5708504692671295626219191080155384660923188, 22391245248814145952527374378611036658055358, 87850424995677427097887635240027432741630786, 344759642971574585017101869284415200820110318, 1353297310861350955121099816118577972404806335, 5313389618538836978256606753183863428572263654, 20866483038251451780385190101227950722276864425, 81964084694416849138775868521619707027102156261, 322027049691979694283467167319603635133233767061, 1265474209338272981070932299759764404267503515689, 4973982704975227175787231796580140912919656966098, 19554341090975715669407647104209113461027374091063, 76889676417104127413812136472738409615045191101896, 302396574748454441151127456068983919860507597576118, 1189509215938458103415523657407957715323107338271180, 4679926667751762879958282643395354779990923914242514, 18415725022809134006915524008410020481320026130170425, 72479540754813931278437619329689195739843446561536996, 285310159923931193693002119775261293963790338430607171, 1123291938708686127998838606028785420246044021143222149, 4423235939159856984803422205195682691305900787322777983, 17420400418800096696653005258366495791188762237669862433, 68619141002624276668517660491245495240750217323562577326, 270333665438335695758320982865433218030749751544364487123, 1065175918782420786686132633666661011466919444729278931410

Generating function in Maple syntax:
1/2*((-x^5+8*x^4-16*x^3+14*x^2-6*x+1)*(1-4*x)^(1/2)+3*x^5-14*x^4+18*x^3-14*x^2+6*x-1)/x/(x^2-3*x+1)/(x-1)^3

Generating function in latex syntax:
\frac{\left(-x^{5}+8 x^{4}-16 x^{3}+14 x^{2}-6 x +1\right) \sqrt{1-4 x}+3 x^{5}-14 x^{4}+18 x^{3}-14 x^{2}+6 x -1}{2 x \left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}

Generating function in sympy syntax:
(3*x**5/2 - 7*x**4 + 9*x**3 - 7*x**2 + 3*x + sqrt(1 - 4*x)*(-x**5 + 8*x**4 - 16*x**3 + 14*x**2 - 6*x + 1)/2 - 1/2)/(x*(x - 1)**3*(x**2 - 3*x + 1))

Implicit equation for the generating function in Maple syntax:
x*(x^2-3*x+1)^2*(x-1)^6*F(x)^2-(3*x^5-14*x^4+18*x^3-14*x^2+6*x-1)*(x^2-3*x+1)*(x-1)^3*F(x)+x^10-14*x^9+79*x^8-232*x^7+416*x^6-481*x^5+371*x^4-191*x^3+63*x^2-12*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)^{2}-\left(3 x^{5}-14 x^{4}+18 x^{3}-14 x^{2}+6 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{10}-14 x^{9}+79 x^{8}-232 x^{7}+416 x^{6}-481 x^{5}+371 x^{4}-191 x^{3}+63 x^{2}-12 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 214
a(7) = 691
a(8) = 2247
a(9) = 7403
a(10) = 24766
a(n+8) = -2*(2*n+1)/(n+9)*a(n)+(76+45*n)/(n+9)*a(n+1)-5*(35*n+89)/(n+9)*a(n+2)+(1100+321*n)/(n+9)*a(n+3)-(326*n+1455)/(n+9)*a(n+4)+(1093+196*n)/(n+9)*a(n+5)-(69*n+463)/(n+9)*a(n+6)+(102+13*n)/(n+9)*a(n+7)+1/2*(3*n^2-3*n+2)/(n+9), n >= 11

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) = 66
a \! \left(6\right) = 214
a \! \left(7\right) = 691
a \! \left(8\right) = 2247
a \! \left(9\right) = 7403
a \! \left(10\right) = 24766
a \! \left(n +8\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +9}+\frac{\left(76+45 n \right) a \! \left(n +1\right)}{n +9}-\frac{5 \left(35 n +89\right) a \! \left(n +2\right)}{n +9}+\frac{\left(1100+321 n \right) a \! \left(n +3\right)}{n +9}-\frac{\left(326 n +1455\right) a \! \left(n +4\right)}{n +9}+\frac{\left(1093+196 n \right) a \! \left(n +5\right)}{n +9}-\frac{\left(69 n +463\right) a \! \left(n +6\right)}{n +9}+\frac{\left(102+13 n \right) a \! \left(n +7\right)}{n +9}+\frac{3 n^{2}-3 n +2}{2 n +18}, \quad n \geq 11

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21378/
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[5,x]
F[4,x] = x
F[5,x] = F[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[4,x]*F[8,x]
F[8,x] = F[28,x]+F[9,x]
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[13,x]+F[20,x]
F[13,x] = F[14,x]*F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]^2*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[21,x]+F[23,x]+F[24,x]
F[23,x] = 0
F[24,x] = F[25,x]
F[25,x] = F[16,x]*F[26,x]*F[4,x]
F[26,x] = F[16,x]+F[27,x]
F[27,x] = F[22,x]
F[28,x] = F[29,x]
F[29,x] = F[0,x]*F[17,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{28}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{20}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17} \left(x \right)^{2} F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{21}\! \left(x \right)+F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right) F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{0}\! \left(x \right) F_{17}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_4(x)*F_8(x))
Eq(F_8(x), F_28(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_13(x) + F_20(x))
Eq(F_13(x), F_14(x)*F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)**2*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_21(x) + F_23(x) + F_24(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_16(x)*F_26(x)*F_4(x))
Eq(F_26(x), F_16(x) + F_27(x))
Eq(F_27(x), F_22(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_0(x)*F_17(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, 2, 1, 3], "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": [2, 1, 3, 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, 2, 1, 3], "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": [2, 1, 3, 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, 2, 1, 3], "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": [2, 1, 3, 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"}}, {"children": [{"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]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 2], [1, 0], [1, 2]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": 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