0213_0231_1032_1230_1302_3120

Counting sequence:
1, 1, 2, 6, 18, 46, 108, 240, 514, 1074, 2208, 4492, 9078, 18270, 36676, 73512, 147210, 294634, 589512, 1179300, 2358910, 4718166, 9436716, 18873856, 37748178, 75496866, 150994288, 301989180, 603979014, 1207958734, 2415918228, 4831837272, 9663675418, 19327351770, 38654704536, 77309410132, 154618821390, 309237643974, 618475289212, 1236950579760, 2473901160930, 4947802323346, 9895604648256, 19791209298156, 39582418598038, 79164837197886, 158329674397668, 316659348797320, 633318697596714, 1266637395195594, 2533274790393448, 5066549580789252, 10133099161580958, 20266198323164470, 40532396646331596, 81064793292665952, 162129586585334770, 324259173170672514, 648518346341348112, 1297036692682699420, 2594073385365402150, 5188146770730807726, 10376293541461618996, 20752587082923241656, 41505174165846487098, 83010348331692978106, 166020696663385960248, 332041393326771924660, 664082786653543853614, 1328165573307087711654, 2656331146614175427868, 5312662293228350860432, 10625324586456701725698, 21250649172913403456370, 42501298345826806917856, 85002596691653613840972, 170005193383307227687350, 340010386766614455380254, 680020773533228910766212, 1360041547066457821538280, 2720083094132915643082570, 5440166188265831286171306, 10880332376531662572348936, 21760664753063325144704356, 43521329506126650289415358, 87042659012253300578837526, 174085318024506601157682028, 348170636049013202315371200, 696341272098026404630749714, 1392682544196052809261506914, 2785365088392105618523021488, 5570730176784211237046050812, 11141460353568422474092109638, 22282920707136844948184227470, 44565841414273689896368463316, 89131682828547379792736935192, 178263365657094759585473879130, 356526731314189519170947767194, 713053462628379038341895543512, 1426106925256758076683791096340, 2852213850513516153367582202190

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

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

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

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

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

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,-6-n^2+n+9/4*2^n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ -6-n^{2}+n +\frac{9 \,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) = 46
a(6) = 108
a(n+1) = n^2+2*a(n)-3*n+6, 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) = 46
a \! \left(6\right) = 108
a \! \left(n +1\right) = n^{2}+2 a \! \left(n \right)-3 n +6, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22530/
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[17,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[17,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]*F[17,x]
F[10,x] = F[11,x]+F[18,x]+F[9,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]*F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[17,x]
F[17,x] = x
F[18,x] = F[19,x]
F[19,x] = F[14,x]*F[17,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[17,x]*F[23,x]
F[23,x] = F[24,x]+F[40,x]
F[24,x] = F[25,x]+F[38,x]
F[25,x] = F[26,x]+F[30,x]
F[26,x] = F[20,x]*F[27,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[14,x]^2*F[17,x]
F[30,x] = F[31,x]
F[31,x] = F[15,x]*F[17,x]*F[32,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[15,x]*F[34,x]
F[34,x] = F[1,x]+F[35,x]
F[35,x] = F[17,x]*F[34,x]
F[36,x] = F[37,x]
F[37,x] = F[17,x]*F[32,x]*F[34,x]
F[38,x] = F[14,x]*F[34,x]*F[39,x]
F[39,x] = F[12,x]
F[40,x] = F[41,x]*F[6,x]
F[41,x] = F[14,x]+F[15,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_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \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_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \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_{18}\! \left(x \right)+F_{9}\! \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_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = x
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{17}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{40}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{38}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \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)^{2} F_{17}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right) F_{34}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{17}\! \left(x \right) F_{32}\! \left(x \right) F_{34}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right) F_{34}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right) F_{6}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \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_17(x)*F_4(x))
Eq(F_4(x), F_21(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_17(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_17(x))
Eq(F_10(x), F_11(x) + F_18(x) + F_9(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x)*F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_17(x))
Eq(F_17(x), x)
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_14(x)*F_17(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_17(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_40(x))
Eq(F_24(x), F_25(x) + F_38(x))
Eq(F_25(x), F_26(x) + F_30(x))
Eq(F_26(x), F_20(x)*F_27(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_14(x)**2*F_17(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_15(x)*F_17(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_15(x)*F_34(x))
Eq(F_34(x), F_1(x) + F_35(x))
Eq(F_35(x), F_17(x)*F_34(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_17(x)*F_32(x)*F_34(x))
Eq(F_38(x), F_14(x)*F_34(x)*F_39(x))
Eq(F_39(x), F_12(x))
Eq(F_40(x), F_41(x)*F_6(x))
Eq(F_41(x), F_14(x) + F_15(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, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 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], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": 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