0132_0231_0321_2301_3120

Counting sequence:
1, 1, 2, 6, 19, 51, 119, 254, 515, 1017, 1987, 3876, 7583, 14903, 29423, 58314, 115915, 230901, 460619, 919760, 1837703, 3673203, 7343767, 14684406, 29365139, 58726001, 117447059, 234888444, 469770415, 939533487, 1879058687, 3758108066, 7516205723, 15032399853, 30064786843, 60129559464, 120259103255, 240518189291, 481036359719, 962072698830, 1924145375203, 3848290725993, 7696581425507, 15393162822356, 30786325613759, 61572651194151, 123145302352399, 246290604666234, 492581209291115, 985162418537957, 1970324837028587, 3940649674006656, 7881299347959463, 15762598695861603, 31525197391662263, 63050394783259814, 126100789566450995, 252201579132829281, 504403158265581619, 1008806316531081900, 2017612633062077903, 4035225266124065183, 8070450532248034847, 16140901064495969106, 32281802128991832379, 64563604257983553501, 129127208515966990139, 258254417031933857624, 516508834063867586615, 1033017668127735038427, 2066035336255469935687, 4132070672510939723646, 8264141345021879292803, 16528282690043758424153, 33056565380087516679683, 66113130760175033183364, 132226261520350066183135, 264452523040700132174871, 528905046081400264150319, 1057810092162800528092970, 2115620184325601055969803, 4231240368651202111714773, 8462480737302404223195787, 16924961474604808446148656, 33849922949209616892044999, 67699845898419233783828051, 135399691796838467567384279, 270799383593676935134486614, 541598767187353870268680915, 1083197534374707740537058897, 2166395068749415481073803987, 4332790137498830962147283036, 8665580274997661924294229743, 17331160549995323848588111503, 34662321099990647697175863103, 69324642199981295394351354114, 138649284399962590788702323675, 277298568799925181577404250061, 554597137599850363154808089819, 1109194275199700726309615756040, 2218388550399401452619231074903

Generating function in Maple syntax:
(x^6+4*x^5-4*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}+4 x^{5}-4 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 + 4*x**5 - 4*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+4*x^5-4*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}+4 x^{5}-4 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,-5+1/2*n^3-7/2*n^2+5*n+7/4*2^n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -5+\frac{n^{3}}{2}-\frac{7 n^{2}}{2}+5 n +\frac{7 \,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) = 19
a(5) = 51
a(6) = 119
a(n+1) = -1/2*n^3+5*n^2+2*a(n)-21/2*n+7, 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) = 19
a \! \left(5\right) = 51
a \! \left(6\right) = 119
a \! \left(n +1\right) = -\frac{n^{3}}{2}+5 n^{2}+2 a \! \left(n \right)-\frac{21 n}{2}+7, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21687/
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[14,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[24,x]+F[8,x]
F[8,x] = F[23,x]+F[9,x]
F[9,x] = F[10,x]+F[17,x]
F[10,x] = F[11,x]*F[15,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[9,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[14,x]*F[15,x]
F[17,x] = F[18,x]*F[21,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[14,x]^2*F[15,x]^2
F[23,x] = F[15,x]*F[18,x]*F[19,x]
F[24,x] = F[25,x]+F[27,x]
F[25,x] = F[15,x]*F[19,x]*F[26,x]
F[26,x] = F[18,x]+F[19,x]
F[27,x] = F[18,x]*F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[14,x]*F[19,x]*F[30,x]*F[31,x]
F[30,x] = F[1,x]+F[14,x]
F[31,x] = F[15,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[14,x]*F[15,x]*F[31,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_{14}\! \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_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{24}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{23}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14} \left(x \right)^{2} F_{15} \left(x \right)^{2}
F_{23}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right) F_{30}\! \left(x \right) F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{31}\! \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_14(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_14(x)*F_7(x))
Eq(F_7(x), F_24(x) + F_8(x))
Eq(F_8(x), F_23(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_17(x))
Eq(F_10(x), F_11(x)*F_15(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_9(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_14(x)*F_15(x))
Eq(F_17(x), F_18(x)*F_21(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_14(x)**2*F_15(x)**2)
Eq(F_23(x), F_15(x)*F_18(x)*F_19(x))
Eq(F_24(x), F_25(x) + F_27(x))
Eq(F_25(x), F_15(x)*F_19(x)*F_26(x))
Eq(F_26(x), F_18(x) + F_19(x))
Eq(F_27(x), F_18(x)*F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_14(x)*F_19(x)*F_30(x)*F_31(x))
Eq(F_30(x), F_1(x) + F_14(x))
Eq(F_31(x), F_15(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_14(x)*F_15(x)*F_31(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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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": []}, "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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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": [[{"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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 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, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"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], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [2, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [2, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, 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