0123_0132_0231_0312_1203_3012

Counting sequence:
1, 1, 2, 6, 18, 53, 159, 495, 1600, 5333, 18194, 63167, 222294, 790797, 2838452, 10265484, 37368348, 136803359, 503347857, 1860283302, 6902856644, 25706763213, 96048126849, 359937671342, 1352548393986, 5095293095211, 19239398084785, 72802194266612, 276033039859496, 1048530949011737, 3989799629112615, 15206242263066231, 58042975116516968, 221867842088244069, 849218688722393562, 3254557274713803747, 12487657627650216622, 47968785399293917861, 184458785560367742248, 710033854453692613065, 2735746507526853342162, 10550410324778199138269, 40723080570027955885026, 157315702487561837767473, 608201247790833814769270, 2353157649393357935862117, 9111060415375553450595294, 35301050730560691295020491, 136865691700144516457980098, 530979855048809211219395457, 2061232157720372111473191008, 8006280032499519055742685255, 31115718386884408799349170610, 120994274850882546898111688885, 470735553922417979898405464400, 1832348632493796467201543382495, 7135925742667295307722572807838, 27803292807509602526035178888073, 108377422329353086935550993090224, 422640787138732377879154165083155, 1648874747316403472952182100240666, 6435489353041592851773327638763741, 25127391365393720516653125122393372, 98147601989762452729853826400338756, 383506952654785390083712694985826404, 1499074058560248162377498651821221191, 5861712294441154476424684848164713017, 22928357648025842597219546463257412582, 89714803806085984065853813436177013836, 351150956347826249650630015073049665829, 1374859188159761840349695870190032364561, 5384599606157799430782516421576678066682, 21094819151373869017091096187073119282642, 82665045793695465529953497676249898968891, 324032360918121976172068295215632267576269, 1270492252116954382883212049153802760467576, 4982757321395670124918097676418166848375992, 19546930467765378478989209996935562733851057, 76700053913418795913137775653275277459551811, 301035906179453100228211268948248726151613804, 1181799552522007240372061484361993051402481086, 4640552552613187581375882626475111368502143267, 18226088505946818326379989951698862133350712227, 71599963648417725126456737833473325130243655446, 281336060667930772390789855010708092260085337904, 1105679052766915026130152765763379864412920941101, 4346320257853259067026058713999472423110205407117, 17088395398689636996522675150099578912602585401492, 67199460777712256997453292162433944740755078943366, 264309837445629452873488872953694331036173223461159, 1039781433048082360093658258182336205289720598608871, 4091195153680013707859477591338608425447687730637702, 16100375923482363192833910869520431287289293977164252, 63372063511071892401348101500512472516064569902191121, 249479047295980027469315116256637538138186949655140709, 982297718307425207996981186464894002647953895728542266, 3868329923573308211497146572010139904262022313261066742, 15236096799937625941852895106776486816076519335721098487, 60019497961830873026892905482973432766101831827302066101, 236471078063067052092924249812132360993456312367974449224, 931814366019893684188221478805415911052914629566130848156

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

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

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

Implicit equation for the generating function in Maple syntax:
x*(x-1)^6*F(x)^2+(3*x^2-3*x+1)*(x-1)^3*F(x)+(3*x^2-3*x+1)^2 = 0

Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{6} F \! \left(x \right)^{2}+\left(3 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(3 x^{2}-3 x +1\right)^{2} = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(n+4) = -6*(3+2*n)/(5+n)*a(n)+9*(5+3*n)/(5+n)*a(1+n)-2*(29+11*n)/(5+n)*a(n+2)+2*(15+4*n)/(5+n)*a(n+3)+(n-1)/(5+n), n >= 5

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(n +4\right) = -\frac{6 \left(3+2 n \right) a \! \left(n \right)}{5+n}+\frac{9 \left(5+3 n \right) a \! \left(1+n \right)}{5+n}-\frac{2 \left(29+11 n \right) a \! \left(n +2\right)}{5+n}+\frac{2 \left(15+4 n \right) a \! \left(n +3\right)}{5+n}+\frac{n -1}{5+n}, \quad n \geq 5

Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22074/
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[17,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[17,x]*F[7,x]
F[7,x] = F[18,x]+F[8,x]
F[8,x] = F[9,x,1]
F[9,x,k[0]] = F[10,x,k[0]]+F[14,x,k[0]]
F[10,x,k[0]] = F[0,x]+F[11,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]*F[9,x,k[0]]
F[13,x,k[0]] = k[0]*x
F[14,x,k[0]] = F[15,x,k[0]]
F[15,x,k[0]] = F[16,x,k[0]]*F[17,x]
F[16,x,k[0]] = (F[9,x,k[0]]*k[0]-F[9,x,1])/(-1+k[0])
F[17,x] = x
F[18,x] = F[19,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[17,x]*F[22,x]
F[22,x] = F[1,x]+F[20,x]
F[23,x] = F[24,x]+F[25,x]+F[28,x]
F[24,x] = 0
F[25,x] = F[17,x]*F[26,x]
F[26,x] = F[20,x]+F[27,x]
F[27,x] = F[25,x]
F[28,x] = F[17,x]*F[19,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_{17}\! \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_{17}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{18}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x , 1\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = y x
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = F_{16}\! \left(x , y\right) F_{17}\! \left(x \right)
F_{16}\! \left(x , y\right) = \frac{y F_{9}\! \left(x , y\right)-F_{9}\! \left(x , 1\right)}{-1+y}
F_{17}\! \left(x \right) = x
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{17}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right) F_{19}\! \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_17(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_17(x)*F_7(x))
Eq(F_7(x), F_18(x) + F_8(x))
Eq(F_8(x), F_9(x, 1))
Eq(F_9(x, k_0), F_10(x, k_0) + F_14(x, k_0))
Eq(F_10(x, k_0), F_0(x) + F_11(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0)*F_9(x, k_0))
Eq(F_13(x, k_0), k_0*x)
Eq(F_14(x, k_0), F_15(x, k_0))
Eq(F_15(x, k_0), F_16(x, k_0)*F_17(x))
Eq(F_16(x, k_0), (-k_0*F_9(x, k_0) + F_9(x, 1))/(1 - k_0))
Eq(F_17(x), x)
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_17(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_20(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_28(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_17(x)*F_26(x))
Eq(F_26(x), F_20(x) + F_27(x))
Eq(F_27(x), F_25(x))
Eq(F_28(x), F_17(x)*F_19(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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "point_placements_tracked_fusion", "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 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, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 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, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 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, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 0]]}, {"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, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 1, 2], "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, 0]]}, {"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, 0], [0, 2], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 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