0132_0231_1302_1320_3021

Counting sequence:
1, 1, 2, 6, 19, 58, 175, 537, 1699, 5552, 18660, 64135, 224275, 794814, 2846552, 10281762, 37400995, 136868758, 503478775, 1860545274, 6903380741, 25707811578, 96050223769, 359941865392, 1352556782317, 5095309872126, 19239431638891, 72802261375124, 276033174076845, 1048531217446786, 3989800165983091, 15206243336807589, 58042977264000119, 221867846383210836, 849218697312327592, 3254557291893672335, 12487657662009954359, 47968785468013393930, 184458785697806695016, 710033854728570519267, 2735746508076609155269, 10550410325877710765224, 40723080572226979139716, 157315702491959884277673, 608201247799629907790531, 2353157649410950121905542, 9111060415410737822683090, 35301050730631060039197073, 136865691700285253946334297, 530979855049090686196104936, 2061232157720935061426611094, 8006280032500644955649526603, 31115718386886660599162854531, 120994274850887050497739058002, 470735553922426987097660203960, 1832348632493814481600052862993, 7135925742667331336519591770265, 27803292807509674583629216814412, 108377422329353231050739068944442, 422640787138732666109530316793187, 1648874747316404049412934403662383, 6435489353041594004694832245608886, 25127391365393722822496134336085432, 98147601989762457341539844827724706, 383506952654785399307084731840600195, 1499074058560248180824242725530770726, 5861712294441154513318172995583814103, 22928357648025842671006522758095616834, 89714803806085984213427766025853424485, 351150956347826249945777920252402489338, 1374859188159761840939991680548738013857, 5384599606157799431963108042294089367620, 21094819151373869019452279428507941886933, 82665045793695465534675864159119544179958, 324032360918121976181513028181371558000959, 1270492252116954382902101515085281341319584, 4982757321395670124955876608281124010082709, 19546930467765378479064767860661477057267266, 76700053913418795913288891380727106106387079, 301035906179453100228513500403152383445287266, 1181799552522007240372665947271800365989831013, 4640552552613187581377091552294725997676846202, 18226088505946818326382407803338091391700121257, 71599963648417725126461573536751783646942476746, 281336060667930772390799526417265009293482983825, 1105679052766915026130172108576493698479716236346, 4346320257853259067026097399625700091243796001093, 17088395398689636996522752521352034248869766593014, 67199460777712256997453446904938855413289441330065, 264309837445629452873489182438704152381241948238298, 1039781433048082360093658877152355847979858048166977, 4091195153680013707859478829278647710827962629757830, 16100375923482363192833913345400509858049843775408513, 63372063511071892401348106452272629657585669498683738, 249479047295980027469315126160157852421229148848130129, 982297718307425207996981206271934631214038294114525384, 3868329923573308211497146611624221161394191110033037349, 15236096799937625941852895186004649330340856929265044166, 60019497961830873026892905641429757794630507014389962019, 236471078063067052092924250129045011050513662742150245716, 931814366019893684188221479439241211167029330314482445893

Generating function in Maple syntax:
1/2*((6*x^3-9*x^2+5*x-1)*(1-4*x)^(1/2)+2*x^5-6*x^3+9*x^2-5*x+1)/x/(2*x-1)/(x-1)^3

Generating function in latex syntax:
\frac{\left(6 x^{3}-9 x^{2}+5 x -1\right) \sqrt{1-4 x}+2 x^{5}-6 x^{3}+9 x^{2}-5 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{3}}

Generating function in sympy syntax:
(x**5 - 3*x**3 + 9*x**2/2 - 5*x/2 + sqrt(1 - 4*x)*(6*x**3 - 9*x**2 + 5*x - 1)/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-(2*x-1)*(2*x^5-6*x^3+9*x^2-5*x+1)*(x-1)^3*F(x)+x^9-6*x^7+45*x^6-113*x^5+142*x^4-102*x^3+43*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(2 x -1\right) \left(2 x^{5}-6 x^{3}+9 x^{2}-5 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}-6 x^{7}+45 x^{6}-113 x^{5}+142 x^{4}-102 x^{3}+43 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) = 175
a(7) = 537
a(8) = 1699
a(n+6) = -24*(3+2*n)/(7+n)*a(n)+12*(25+13*n)/(7+n)*a(1+n)-2*(281+104*n)/(7+n)*a(n+2)+49*(11+3*n)/(7+n)*a(n+3)-2*(137+29*n)/(7+n)*a(n+4)+2*(35+6*n)/(7+n)*a(n+5)-(n+3)/(7+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) = 19
a \! \left(5\right) = 58
a \! \left(6\right) = 175
a \! \left(7\right) = 537
a \! \left(8\right) = 1699
a \! \left(n +6\right) = -\frac{24 \left(3+2 n \right) a \! \left(n \right)}{7+n}+\frac{12 \left(25+13 n \right) a \! \left(1+n \right)}{7+n}-\frac{2 \left(281+104 n \right) a \! \left(n +2\right)}{7+n}+\frac{49 \left(11+3 n \right) a \! \left(n +3\right)}{7+n}-\frac{2 \left(137+29 n \right) a \! \left(n +4\right)}{7+n}+\frac{2 \left(35+6 n \right) a \! \left(n +5\right)}{7+n}-\frac{n +3}{7+n}, \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21724/
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[17,x]
F[14,x] = F[15,x]+F[24,x]
F[15,x] = F[16,x]*F[19,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[16,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[19,x]*F[22,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[12,x]*F[22,x]
F[24,x] = F[10,x]*F[22,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_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \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_{19}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \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)*F_17(x))
Eq(F_14(x), F_15(x) + F_24(x))
Eq(F_15(x), F_16(x)*F_19(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_16(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_19(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_12(x)*F_22(x))
Eq(F_24(x), F_10(x)*F_22(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, 3, 0, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 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, 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, 3, 0, 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]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [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": [[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, 2], [0, 2], [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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 2], [0, 0], [0, 0], [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}}, {"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"}}, {"children": [{"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, 3, 0, 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, 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