0132_0213_1203_1230_2013_2301

Counting sequence:
1, 1, 2, 6, 18, 48, 117, 268, 588, 1252, 2611, 5368, 10930, 22112, 44545, 89492, 179480, 359564, 719855, 1440576, 2882174, 5765544, 11532477, 23066556, 46134948, 92271988, 184546347, 369095368, 738193738, 1476390832, 2952785401, 5905574948, 11811154480, 23622314012, 47244633575, 94489273232, 188978553110, 377957113464, 755914234805, 1511828478156, 3023656965564, 6047313941124, 12094627893027, 24189255797656, 48378511607778, 96757023228928, 193514046472177, 387028092959668, 774056185935688, 1548112371888812, 3096224743796191, 6192449487612128, 12384898975245230, 24769797950512712, 49539595901049005, 99079191802122972, 198158383604272340, 396316767208572564, 792633534417174555, 1585267068834380136, 3170534137668792954, 6341068275337620304, 12682136550675276777, 25364273101350591556, 50728546202701223008, 101457092405402487868, 202914184810805019607, 405828369621610085168, 811656739243220218438, 1623313478486440487192, 3246626956972881026981, 6493253913945762108908, 12986507827891524275180, 25973015655783048610212, 51946031311566097282835, 103892062623132194630712, 207784125246264389329170, 415568250492528778728864, 831136500985057557531105, 1662273001970115115138516, 3324546003940230230356344, 6649092007880460460795084, 13298184015760920921675727, 26596368031521841843440256, 53192736063043683686972638, 106385472126087367374040808, 212770944252174734748180637, 425541888504349469496463868, 851083777008698938993033988, 1702167554017397877986177972, 3404335108034795755972469771, 6808670216069591511945057288, 13617340432139183023890236330, 27234680864278366047780598512, 54469361728556732095561327065, 108938723457113464191122788452, 217877446914226928382245715600, 435754893828453856764491574364, 871509787656907713528983296455, 1743019575313815427057966745296, 3486039150627630854115933647734

Generating function in Maple syntax:
-(x^6-x^5+3*x^4-8*x^3+10*x^2-5*x+1)/(2*x-1)/(x-1)^4

Generating function in latex syntax:
-\frac{x^{6}-x^{5}+3 x^{4}-8 x^{3}+10 x^{2}-5 x +1}{\left(2 x -1\right) \left(x -1\right)^{4}}

Generating function in sympy syntax:
(-x**6 + x**5 - 3*x**4 + 8*x**3 - 10*x**2 + 5*x - 1)/((x - 1)**4*(2*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^4*F(x)+x^6-x^5+3*x^4-8*x^3+10*x^2-5*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{6}-x^{5}+3 x^{4}-8 x^{3}+10 x^{2}-5 x +1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-23/6*n-1/6*n^3+11/4*2^n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -\frac{23 n}{6}-\frac{n^{3}}{6}+\frac{11 \,2^{n}}{4} & \mathit{\text{otherwise}} \end{array}\right.

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 48
a(6) = 117
a(n+1) = 1/6*n^3-1/2*n^2+2*a(n)+10/3*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) = 18
a \! \left(5\right) = 48
a \! \left(6\right) = 117
a \! \left(n +1\right) = \frac{n^{3}}{6}-\frac{n^{2}}{2}+2 a \! \left(n \right)+\frac{10 n}{3}-4, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22257/
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[30,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[12,x]*F[7,x]
F[7,x] = F[5,x]+F[8,x]
F[8,x] = F[12,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]+F[8,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[10,x]*F[12,x]
F[12,x] = x
F[13,x] = F[14,x]
F[14,x] = F[10,x]*F[12,x]*F[15,x]
F[15,x] = F[16,x]+F[25,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[18,x]+F[20,x]
F[18,x] = F[19,x]
F[19,x] = F[10,x]*F[12,x]
F[20,x] = F[21,x]+F[22,x]+F[24,x]
F[21,x] = 0
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[18,x]+F[20,x]
F[24,x] = F[12,x]*F[17,x]
F[25,x] = F[18,x]+F[26,x]
F[26,x] = F[21,x]+F[27,x]+F[28,x]
F[27,x] = F[12,x]*F[25,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[18,x]+F[26,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[40,x]
F[33,x] = F[10,x]*F[34,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[10,x]*F[36,x]
F[36,x] = F[1,x]+F[37,x]
F[37,x] = F[12,x]*F[36,x]
F[38,x] = F[39,x]
F[39,x] = F[10,x]^2*F[12,x]^2
F[40,x] = F[41,x]
F[41,x] = F[10,x]*F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[50,x]
F[43,x] = F[44,x]+F[49,x]
F[44,x] = F[16,x]*F[45,x]
F[45,x] = F[1,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[19,x]+F[45,x]
F[49,x] = F[10,x]*F[25,x]
F[50,x] = F[10,x]*F[16,x]*F[51,x]
F[51,x] = F[37,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_{30}\! \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_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{5}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{25}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)+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_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{18}\! \left(x \right)+F_{20}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right)+F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{12} \left(x \right)^{2}
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{50}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{1}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{19}\! \left(x \right)+F_{45}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{50}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{37}\! \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_30(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_12(x)*F_7(x))
Eq(F_7(x), F_5(x) + F_8(x))
Eq(F_8(x), F_12(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x) + F_8(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_10(x)*F_12(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_10(x)*F_12(x)*F_15(x))
Eq(F_15(x), F_16(x) + F_25(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_18(x) + F_20(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_10(x)*F_12(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_24(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_18(x) + F_20(x))
Eq(F_24(x), F_12(x)*F_17(x))
Eq(F_25(x), F_18(x) + F_26(x))
Eq(F_26(x), F_21(x) + F_27(x) + F_28(x))
Eq(F_27(x), F_12(x)*F_25(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_18(x) + F_26(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_40(x))
Eq(F_33(x), F_10(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_10(x)*F_36(x))
Eq(F_36(x), F_1(x) + F_37(x))
Eq(F_37(x), F_12(x)*F_36(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_10(x)**2*F_12(x)**2)
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_10(x)*F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_50(x))
Eq(F_43(x), F_44(x) + F_49(x))
Eq(F_44(x), F_16(x)*F_45(x))
Eq(F_45(x), F_1(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_19(x) + F_45(x))
Eq(F_49(x), F_10(x)*F_25(x))
Eq(F_50(x), F_10(x)*F_16(x)*F_51(x))
Eq(F_51(x), F_37(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": [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]]}, {"patt": [2, 3, 0, 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": [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]]}, {"patt": [2, 3, 0, 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": [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]]}, {"patt": [2, 3, 0, 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": 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