0123_0132_0231_1203_1230_3012

Counting sequence:
1, 1, 2, 6, 18, 52, 152, 464, 1486, 4946, 16916, 58951, 208232, 743186, 2674804, 9695300, 35358230, 129645470, 477639516, 1767264159, 6564121560, 24466268350, 91482565180, 343059615421, 1289904149348, 4861946403752, 18367353074752, 69533550918929, 263747951753636, 1002242216655022, 3814986502096364, 14544636039231404, 55534064877053158, 212336130412248566, 812944042149736748, 3116285494907307807, 11959798385860460632, 45950804324621750134, 176733862787006709836, 680425371729975809529, 2622127042276492118700, 10113918591637898144680, 39044429911904443970720, 150853479205085351673041, 583300119592996693101284, 2257117854077248073267910, 8740328711533173390061500, 33868773757191046886445705, 131327898242169365478009196, 509552245179617138054626996, 1978261657756160653623794056, 7684785670514316385230836981, 29869166945772625950142439612, 116157871455782434250553869306, 451959718027953471447609534228, 1759414616608818870992479902207, 6852456927844873497549658492032, 26700952856774851904245220941924, 104088460289122304033498318842936, 405944995127576985730643443399621, 1583850964596120042686772779073116, 6182127958584855650487080847252326, 24139737743045626825711458546311132, 94295850558771979787935384946419836, 368479169875816659479009042713588614, 1440418573150919668872489894243909030, 5632681584560312734993915705849190860, 22033725021956517463358552614056997855, 86218923998960285726185640663701158616, 337485502510215975556783793455058677094, 1321422108420282270489942177190229599340, 5175569924646105559418940193995065773505, 20276890389709399862928998568254641085340, 79463489365077377841208237632349268946696, 311496878311103321137536291518809134092064, 1221395654430378811828760722007962130858545, 4790408930363303911328386208394864461094820, 18793142726809884575211361279087545193323190, 73745243611532458459690151854647329239411676, 289450081175264899454283846029490767264471309, 1136359577947336271931632877004667456667696100, 4462290049988320482463241297506133183499740060, 17526585015616776834735140517915655636396322840, 68854441132780194707888052034668647142985297981, 270557451039395118028642463289168566420671375724, 1063353702922273835973036658043476458723103503290, 4180080073556524734514695828170907458428751416660, 16435314834665426797069144960762886143367590500935, 64633260585762914370496637486146181462681535370736, 254224158304000796523953440778841647086547372140164, 1000134600800354781929399250536541864362461090068280, 3935312233584004685417853572763349509774031680145285, 15487357822491889407128326963778343232013931127961180, 60960876535340415751462563580829648891969728907567766, 239993345518077005168915776623476723006280827488363644, 944973797977428207852605870454939596837230758235042465, 3721443204405954385563870541379246659709506697378837180, 14657929356129575437016877846657032761712954950899902540, 57743358069601357782187700608042856334020731624756763096, 227508830794229349661819540395688853956041682601541204189, 896519947090131496687170070074100632420837521538746071020

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

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

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

Implicit equation for the generating function in Maple syntax:
x*(x-1)^8*F(x)^2-(3*x^4-4*x^3+6*x^2-4*x+1)*(x-1)^4*F(x)+x^8-6*x^7+24*x^6-50*x^5+66*x^4-55*x^3+28*x^2-8*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{8} F \! \left(x \right)^{2}-\left(3 x^{4}-4 x^{3}+6 x^{2}-4 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{8}-6 x^{7}+24 x^{6}-50 x^{5}+66 x^{4}-55 x^{3}+28 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) = 52
a(1+n) = 2*(2*n+1)/(n+2)*a(n)-1/2*n*(n-1)*(n^2-3*n-2)/(n+2), n >= 6

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) = 52
a \! \left(1+n \right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{n \left(n -1\right) \left(n^{2}-3 n -2\right)}{2 \left(n +2\right)}, \quad n \geq 6

Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22086/
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[11,x]*F[4,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[11,x]*F[7,x]
F[7,x] = F[8,x,1]
F[8,x,k[0]] = F[1,x]+F[12,x,k[0]]+F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]*F[11,x]
F[10,x,k[0]] = 1/(-1+k[0])*(F[8,x,k[0]]*k[0]-F[8,x,1])
F[11,x] = x
F[12,x,k[0]] = F[13,x,k[0]]*F[8,x,k[0]]
F[13,x,k[0]] = k[0]*x
F[14,x] = F[15,x]
F[15,x] = F[11,x]*F[16,x]
F[16,x] = F[17,x]+F[25,x]
F[17,x] = F[18,x,1]
F[18,x,k[0]] = F[19,x,k[0]]+F[22,x,k[0]]
F[19,x,k[0]] = F[20,x,k[0]]+F[5,x]
F[20,x,k[0]] = F[21,x,k[0]]
F[21,x,k[0]] = F[13,x,k[0]]*F[18,x,k[0]]
F[22,x,k[0]] = F[23,x,k[0]]
F[23,x,k[0]] = F[11,x]*F[24,x,k[0]]
F[24,x,k[0]] = (F[18,x,k[0]]*k[0]-F[18,x,1])/(-1+k[0])
F[25,x] = F[26,x]
F[26,x] = F[27,x]^4*F[11,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[11,x]*F[27,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_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{11}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x , 1\right)
F_{8}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{11}\! \left(x \right)
F_{10}\! \left(x , y\right) = \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}
F_{11}\! \left(x \right) = x
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = y x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{25}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x , 1\right)
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)+F_{22}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{20}\! \left(x , y\right)+F_{5}\! \left(x \right)
F_{20}\! \left(x , y\right) = F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{18}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{23}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = \frac{y F_{18}\! \left(x , y\right)-F_{18}\! \left(x , 1\right)}{-1+y}
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27} \left(x \right)^{4} F_{11}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{27}\! \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_11(x)*F_4(x))
Eq(F_4(x), F_14(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_11(x)*F_7(x))
Eq(F_7(x), F_8(x, 1))
Eq(F_8(x, k_0), F_1(x) + F_12(x, k_0) + F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0)*F_11(x))
Eq(F_10(x, k_0), (k_0*F_8(x, k_0) - F_8(x, 1))/(k_0 - 1))
Eq(F_11(x), x)
Eq(F_12(x, k_0), F_13(x, k_0)*F_8(x, k_0))
Eq(F_13(x, k_0), k_0*x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_11(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_25(x))
Eq(F_17(x), F_18(x, 1))
Eq(F_18(x, k_0), F_19(x, k_0) + F_22(x, k_0))
Eq(F_19(x, k_0), F_20(x, k_0) + F_5(x))
Eq(F_20(x, k_0), F_21(x, k_0))
Eq(F_21(x, k_0), F_13(x, k_0)*F_18(x, k_0))
Eq(F_22(x, k_0), F_23(x, k_0))
Eq(F_23(x, k_0), F_11(x)*F_24(x, k_0))
Eq(F_24(x, k_0), (-k_0*F_18(x, k_0) + F_18(x, 1))/(1 - k_0))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_11(x)*F_27(x)**4)
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_11(x)*F_27(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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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"}}, {"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], [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, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [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, 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, 1, 2], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [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", 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