0132_0213_0231_0312_1023_1203_1230_2013

Counting sequence:
1, 1, 2, 6, 16, 46, 141, 452, 1495, 5059, 17422, 60842, 214930, 766614, 2756940, 9985357, 36391082, 133352642, 491041397, 1816023557, 6742525577, 25122310877, 93905870687, 352047041117, 1323357841811, 4986882659579, 18835348943929, 71291450935105, 270367798170913, 1027229416144073, 3909507252501013, 14902904741385978, 56894575795858635, 217511627833145851, 832664175609860414, 3191539693244479110, 12247388714609874450, 47051338695520894086, 180950682652813154384, 696601989981642706866, 2684254464852780757660, 10352779877001193484260, 39963716568997715109870, 154394892904455114920150, 596955451884004354481510, 2309817616280160178606430, 8943881952941170847059730, 34655627118192041036530940, 134371869457247607701347070, 521336545106228553668010062, 2023914731439963116123605436, 7861766642440288213208639036, 30555700162878214916174848964, 118822666330644183870210051788, 462309298573329537452496531488, 1799633364099967562947509315548, 7008833546791804623755241749768, 27309289193749736501898413707544, 106456211242706007502143991482932, 415165203009005562696838774502276, 1619772125334323471557213331086724, 6322137579612181383391019718076244, 24685692359200529544346040860500332, 96425656139523002549256740039646041, 376791103415152537890817478654029202, 1472870244433301174110009788730620914, 5759429106401466220019370985282280789, 22528951713673487607863016936203632589, 88154569263828175539562594880091656489, 345053829359644178105154663377298317789, 1351024160291666990501671599726526187639, 5291390900516450565156855257195063467889, 20730196868089960844223697424909697401939, 81238217951878221092992878666194554805339, 318447176822550269948739872746649485664429, 1248622843331535160502893301804057123453909, 4897099608629537637843727025823308722571909, 19211330283387221622864238388034798588824669, 75384826822540174319171789685602544765701249, 295880073316635919225093870068840847251782399, 1161582712815517176277654262323105081848254149, 4561258428468033845268952834679218137919630549, 17915003475271755096995135901118231358080602319, 70379219170382509242079680294875890321337188079, 276544519126985753045297332846881942782522917159, 1086867356025481247824426668119105490721351275519, 4272448167792512341073973890281205137569984391419, 16798240379940809521657065486162352388929494752479, 66059539833960570931057594669589124166966543023059, 259830517625754977819029093790455497249261131102979, 1022176274956774390021544048508918600668542439422119, 4021987168326187207880495008221872427542794564756119, 15828256915538072726114921840015707796869241384594519, 62301910229186953852378557707603555321187500254147719, 245269691445507547955249624323013978945272630514963119, 965737501727350640046068045118255196008236492389473169, 3803167784691217233508795279623391907772482160440701419, 14979647281932915290130718361524654732782211241449991819, 59010049793382125843154146993365417901927218673541751769, 232496965722416072108349857322390662183657676347704882409, 916166011374447794570717264847459473410166470235809499489

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

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

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

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

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

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 16
a(5) = 46
a(6) = 141
a(n+5) = -2*(-3+2*n)/(6+n)*a(n)+(-6+5*n)/(6+n)*a(1+n)+(10+3*n)/(6+n)*a(n+2)-(32+9*n)/(6+n)*a(n+3)+2*(14+3*n)/(6+n)*a(n+4), n >= 7

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) = 16
a \! \left(5\right) = 46
a \! \left(6\right) = 141
a \! \left(n +5\right) = -\frac{2 \left(-3+2 n \right) a \! \left(n \right)}{6+n}+\frac{\left(-6+5 n \right) a \! \left(1+n \right)}{6+n}+\frac{\left(10+3 n \right) a \! \left(n +2\right)}{6+n}-\frac{\left(32+9 n \right) a \! \left(n +3\right)}{6+n}+\frac{2 \left(14+3 n \right) a \! \left(n +4\right)}{6+n}, \quad n \geq 7

Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/23298/
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[13,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[13,x]*F[7,x]
F[7,x] = F[8,x,1]
F[8,x,k[0]] = F[1,x]+F[11,x,k[0]]+F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]*F[8,x,k[0]]
F[10,x,k[0]] = k[0]*x
F[11,x,k[0]] = F[12,x,k[0]]*F[13,x]
F[12,x,k[0]] = (F[8,x,k[0]]*k[0]-F[8,x,1])/(-1+k[0])
F[13,x] = F[10,x,1]
F[14,x] = F[15,x]
F[15,x] = F[13,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[10,x,k[0]]*F[18,x,k[0]]
F[22,x,k[0]] = F[23,x,k[0]]
F[23,x,k[0]] = F[13,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[5,x]
F[26,x] = F[27,x]
F[27,x] = F[13,x]*F[28,x]
F[28,x] = F[1,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_{13}\! \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_{13}\! \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_{11}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = y x
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{13}\! \left(x \right)
F_{12}\! \left(x , y\right) = \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}
F_{13}\! \left(x \right) = F_{10}\! \left(x , 1\right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \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_{10}\! \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_{13}\! \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_{5}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \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_13(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_13(x)*F_7(x))
Eq(F_7(x), F_8(x, 1))
Eq(F_8(x, k_0), F_1(x) + F_11(x, k_0) + F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0)*F_8(x, k_0))
Eq(F_10(x, k_0), k_0*x)
Eq(F_11(x, k_0), F_12(x, k_0)*F_13(x))
Eq(F_12(x, k_0), (-k_0*F_8(x, k_0) + F_8(x, 1))/(1 - k_0))
Eq(F_13(x), F_10(x, 1))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(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_10(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_13(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)*F_5(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_13(x)*F_28(x))
Eq(F_28(x), F_1(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, 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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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": [2, 0, 1, 3], "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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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": [2, 0, 1, 3], "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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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": [2, 0, 1, 3], "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], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[1, 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, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[2, 1], [2, 1], [2, 1]]}], 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