0132_0231_0312_0321_1032_1320_3021_3120

Counting sequence:
1, 1, 2, 6, 16, 41, 101, 241, 561, 1281, 2881, 6401, 14081, 30721, 66561, 143361, 307201, 655361, 1392641, 2949121, 6225921, 13107201, 27525121, 57671681, 120586241, 251658241, 524288001, 1090519041, 2264924161, 4697620481, 9730785281, 20132659201, 41607495681, 85899345921, 177167400961, 365072220161, 751619276801, 1546188226561, 3178275799041, 6528350289921, 13400297963521, 27487790694401, 56349970923521, 115448720916481, 236394999971841, 483785116221441, 989560464998401, 2023101395107841, 4134163720437761, 8444249301319681, 17240342323527681, 35184372088832001, 71776119061217281, 146366987889541121, 298363475313295361, 607985949695016961, 1238489897526886401, 2522015791327477761, 5134103575202365441, 10448351135499550721, 21256990241188741121, 43234556422756761601, 87910264726272081921, 178702833214061281281, 363170273951156797441, 737869762948382064641, 1498797955988901068801, 3043712772162076016641, 6179659264692699791361, 12543785970122495098881, 25456506821719181230081, 51650883406386744524801, 104777506338670253178881, 212506491729134034616321, 430915941561855125749761, 873637799330884364533761, 1770887431076116955136001, 3588998526980930362408961, 7272444383619253629091841, 14733783426553293066731521, 29845356171736157750558721, 60446290980731458735308801, 122403739235981203939000321, 247829793020998980814766081, 501704215140071107503063041, 1015497688476288506753187841, 2055173893344869597000499201, 4158704819474324360989245441, 8414123704517819055954984961, 17021675540173978779862958081, 34430207342624638895631892481, 69634127209802640463075737601, 140815679468712006269775380481, 284726209035637463226798571521, 575642118267701827828092764161, 1163663636928257458405176770561, 2352086074642222522308336025601, 4753689750855860255612637020161, 9606414704854550933217203978241, 19410899815994762710418267832321, 39217940444560847108804255416321

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

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

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

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

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

Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 2,2,1+1/16*(5*n-5)*2^n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 2 & n =2 \\ 1+\frac{\left(5 n -5\right) 2^{n}}{16} & \mathit{\text{otherwise}} \end{array}\right.

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 16
a(5) = 41
a(n+2) = -4*a(n)+4*a(n+1)+1, 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) = 16
a \! \left(5\right) = 41
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+1, \quad n \geq 6

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/23417/
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[15,x]*F[7,x]
F[7,x] = F[10,x]+F[8,x]
F[8,x] = F[14,x]*F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = x
F[13,x] = 2*F[11,x]+F[1,x]
F[14,x] = F[1,x]+F[12,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[12,x]*F[17,x]
F[17,x] = 2*F[16,x]+F[1,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_{15}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{10}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
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_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = 2 F_{11}\! \left(x \right)+F_{1}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{1}\! \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_15(x)*F_7(x))
Eq(F_7(x), F_10(x) + F_8(x))
Eq(F_8(x), F_14(x)*F_9(x))
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_13(x))
Eq(F_12(x), x)
Eq(F_13(x), F_1(x) + 2*F_11(x))
Eq(F_14(x), F_1(x) + F_12(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_1(x) + 2*F_16(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:
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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, 1, 2, 0], "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, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": 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{"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, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 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"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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"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, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 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