0132_0213_0231_1320_3021

Counting sequence:
1, 1, 2, 6, 19, 58, 178, 561, 1821, 6064, 20616, 71268, 249745, 885117, 3166842, 11422495, 41487345, 151598908, 556904516, 2055439508, 7618111637, 28341512509, 105797174886, 396155102992, 1487577689459, 5600353099163, 21134070614668, 79928932402236, 302906067794701, 1150091999655755, 4374434019851114, 16665773291395547, 63591325883477401, 242995288464328020, 929796423130948348, 3562326387281979076, 13664824212759539389, 52477141473327214893, 201746335470278099390, 776401121421140338140, 2990812404445827885091, 11531714687706545486287, 44502149146318230771892, 171882836265550823419284, 664403263625832080563669, 2570177621348004199864859, 9949748365667682336355778, 38544730919537312852301224, 149420141278395070799780743, 579605505009259595584614759, 2249696075657614934161777696, 8737206751496842084361430128, 33952270765493838002027418785, 132008864765739174739554663407, 513530871593639578375753560766, 1998714795195973519099321823262, 7783017003047136265210951866991, 30321487623788196310560287592617, 118181986750993866343678110634868, 460833000908094962255248247923868, 1797715436446278299769480706962949, 7015801897412749150383140146429959, 27390936585297533868190349923737802, 106980376201860649523380248548523731, 417987958905039625775610228464105705, 1633732001768137581253627650691361156, 6387787611670565219235200261457681772, 24984355649327942100713611619517886692, 97752866838263447735889674380472125069, 382587113210778817039550413502657605709, 1497844253820368239181661057949128612974, 5865899717956493472359161748967922475588, 22978972084750457959299383855390584086499, 90043222679805599593367948400616220720103, 352933235330269164504784711623597678721732, 1383731857833787876404465799849888489062020, 5426578215481225361617332655597526311591205, 21286876951476106287618014895500831025171803, 83523120991228205795831845295868765364200946, 327798978661466278701778594191021184707370188, 1286802616531271488886312525182375620792902607, 5052626208331455815826884439891040279014621379, 19843616910250383773394059679690137284093778496, 77950794972035729067669131655723743621879946192, 306276743733522963582939763049194415014856991745, 1203646729443309721789519309598196803054460029991, 4731223577637991624176303594569516713512135842382, 18600955545750634412365397151603581112257462852182, 73144622081489854335603837695998478858890031486319, 287682196130120103251236986159309128022331954868609, 1131683937038356309754462978783623007696711790892164, 4452635064364008537591934470036353443529080453929388, 17522138084852251893405888008860892963886489765593749, 68965749293645145409435539243758097865488877862974183, 271490448857178353891990054913111964185333056527133562, 1068928881190971321050216990615039728087304945220145880, 4209346716351117617128008558322923645215935898837403375, 16578710177997330107997759750999326329563044112052073023, 65306362917742821509713334958106690129430416644439936776, 257292769846641852553887686942473554308574007591807449528, 1013831183264557860253494063775712889459922903144035320137

Generating function in Maple syntax:
1/2*((x-1)^5*(-4*x+1)^(1/2)-2*x^6-5*x^5+11*x^4-12*x^3+10*x^2-5*x+1)/x/(2*x-1)/(x-1)^3

Generating function in latex syntax:
\frac{\left(x -1\right)^{5} \sqrt{-4 x +1}-2 x^{6}-5 x^{5}+11 x^{4}-12 x^{3}+10 x^{2}-5 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{3}}

Generating function in sympy syntax:
(-x**6 - 5*x**5/2 + 11*x**4/2 - 6*x**3 + 5*x**2 - 5*x/2 + sqrt(1 - 4*x)*(x - 1)**5/2 + 1/2)/(x*(x - 1)**3*(2*x - 1))

Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^6*F(x)^2+(x^5+3*x^4-4*x^3+4*x^2-3*x+1)*(2*x-1)^2*(x-1)^3*F(x)+x^11+6*x^10-15*x^9+32*x^8-81*x^7+154*x^6-202*x^5+183*x^4-112*x^3+44*x^2-10*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)^{2}+\left(x^{5}+3 x^{4}-4 x^{3}+4 x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+x^{11}+6 x^{10}-15 x^{9}+32 x^{8}-81 x^{7}+154 x^{6}-202 x^{5}+183 x^{4}-112 x^{3}+44 x^{2}-10 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 58
a(6) = 178
a(7) = 561
a(n+3) = 4*(2*n-1)/(4+n)*a(n)-2*(7*n+10)/(4+n)*a(1+n)+(7*n+19)/(4+n)*a(n+2)+1/2*(15*n^2-77*n+64)/(4+n), n >= 8

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) = 58
a \! \left(6\right) = 178
a \! \left(7\right) = 561
a \! \left(n +3\right) = \frac{4 \left(2 n -1\right) a \! \left(n \right)}{4+n}-\frac{2 \left(7 n +10\right) a \! \left(1+n \right)}{4+n}+\frac{\left(7 n +19\right) a \! \left(n +2\right)}{4+n}+\frac{15 n^{2}-77 n +64}{2 n +8}, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21585/
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[13,x]
F[12,x] = F[5,x]^3
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[8,x]
F[15,x] = F[16,x]+F[31,x]
F[16,x] = F[5,x]^2*F[17,x]
F[17,x] = F[18,x]+F[20,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[18,x]*F[8,x]
F[20,x] = F[21,x]+F[25,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[8,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[23,x]*F[8,x]
F[25,x] = F[26,x]+F[27,x]+F[30,x]
F[26,x] = 0
F[27,x] = F[28,x]*F[8,x]
F[28,x] = F[25,x]+F[29,x]
F[29,x] = F[19,x]
F[30,x] = F[20,x]*F[8,x]
F[31,x] = F[23,x]*F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[18,x]*F[23,x]*F[8,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_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{5} \left(x \right)^{3}
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{31}\! \left(x \right)
F_{16}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{20}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{18}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{23}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{8}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right)
F_{30}\! \left(x \right) = F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right) F_{23}\! \left(x \right) F_{8}\! \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_13(x))
Eq(F_12(x), F_5(x)**3)
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_8(x))
Eq(F_15(x), F_16(x) + F_31(x))
Eq(F_16(x), F_17(x)*F_5(x)**2)
Eq(F_17(x), F_18(x) + F_20(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_18(x)*F_8(x))
Eq(F_20(x), F_21(x) + F_25(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_8(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_23(x)*F_8(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_30(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_28(x)*F_8(x))
Eq(F_28(x), F_25(x) + F_29(x))
Eq(F_29(x), F_19(x))
Eq(F_30(x), F_20(x)*F_8(x))
Eq(F_31(x), F_23(x)*F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_18(x)*F_23(x)*F_8(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": [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": [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": [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], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 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, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 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], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[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": 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