0132_0231_1023_2301

Counting sequence:
1, 1, 2, 6, 20, 61, 163, 388, 845, 1725, 3368, 6394, 11962, 22285, 41657, 78504, 149483, 287677, 558982, 1094878, 2157968, 4273053, 8489279, 16904428, 33713497, 67305853, 134459540, 268729890, 537226742, 1074168877, 2147992885, 4295570896, 8590646039, 17180703357, 34360711634, 68720607046, 137440260460, 274879412029, 549757540379, 1099513600980, 2199025502885, 4398049062205, 8796095909056, 17592189301450, 35184375753074, 70368748288845, 140737492956017, 281474981846392, 562949959140739, 1125899913197629, 2251799820731102, 4503599635165998, 9007199263348616, 18014398518968029, 36028797029398711, 72057594049385788, 144115188088415537, 288230376165456381, 576460752318440876, 1152921504623229682, 2305843009231539502, 4611686018446798957, 9223372036875860333, 18446744073732423072, 36893488147443880751, 73786976294865015037, 147573952589705383594, 295147905179384095894, 590295810358739364836, 1180591620717447609981, 2361183241434861664019, 4722366482869687185700, 9444732965739335485693, 18889465931478629178301, 37778931862957213484888, 75557863725914378840730, 151115727451828706108714, 302231454903657357006733, 604462909807314654962473, 1208925819614629246822984, 2417851639229258426273819, 4835703278458516780677309, 9671406556917033484749110, 19342813113834066887911294, 38685626227668133688998528, 77371252455336267285670429, 154742504910672534473236271, 309485009821345068842304396, 618970019642690137574081033, 1237940039285380275030967933, 2475880078570760549937757636, 4951760157141521099744024002, 9903520314283042199348903270, 19807040628566084398550656173, 39614081257132168796945792165, 79228162514264337593727317872, 158456325028528675187281233991, 316912650057057350374379529085, 633825300114114700748566167170, 1267650600228229401496929062886, 2535301200456458802993644031836

Generating function in Maple syntax:
-(2*x^6-18*x^5+31*x^4-33*x^3+21*x^2-7*x+1)/(2*x-1)/(-1+x)^6

Generating function in latex syntax:
-\frac{2 x^{6}-18 x^{5}+31 x^{4}-33 x^{3}+21 x^{2}-7 x +1}{\left(2 x -1\right) \left(-1+x \right)^{6}}

Generating function in sympy syntax:
(-2*x**6 + 18*x**5 - 31*x**4 + 33*x**3 - 21*x**2 + 7*x - 1)/((x - 1)**6*(2*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(-1+x)^6*F(x)+2*x^6-18*x^5+31*x^4-33*x^3+21*x^2-7*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(-1+x \right)^{6} F \! \left(x \right)+2 x^{6}-18 x^{5}+31 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0

Explicit closed form in Maple syntax:
2^(n+1)-33/20*n-1+1/40*n^5-1/8*n^4+1/8*n^3-3/8*n^2

Explicit closed form in latex syntax:
2^{n +1}-\frac{33 n}{20}-1+\frac{n^{5}}{40}-\frac{n^{4}}{8}+\frac{n^{3}}{8}-\frac{3 n^{2}}{8}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 61
a(6) = 163
a(n+1) = 2*a(n)-1/40*(n-1)*(n^4-9*n^3+6*n^2-4*n-40), 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) = 20
a \! \left(5\right) = 61
a \! \left(6\right) = 163
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-9 n^{3}+6 n^{2}-4 n -40\right)}{40}, \quad n \geq 7

Specification 1
Strategy pack name: point_and_row_and_col_placements
Tree: http://permpal.com/tree/21221/
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[7,x]
F[7,x] = F[29,x]+F[8,x]
F[8,x] = F[13,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[9,x]
F[12,x] = x
F[13,x] = F[5,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]*F[9,x]
F[16,x] = F[17,x]+F[24,x]
F[17,x] = F[18,x]+F[9,x]
F[18,x] = F[10,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[23,x]
F[20,x] = 0
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[10,x]+F[19,x]
F[23,x] = F[12,x]*F[18,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[20,x]+F[26,x]+F[27,x]
F[26,x] = F[12,x]*F[24,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[10,x]+F[25,x]
F[29,x] = F[30,x]
F[30,x] = F[9,x]^2*F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[9,x]^2
F[33,x] = F[34,x]*F[35,x]
F[34,x] = F[10,x]+F[9,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[1,x]+F[36,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_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{29}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{5}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{24}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{9}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right)+F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{9} \left(x \right)^{2}
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right)+F_{9}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{1}\! \left(x \right)+F_{36}\! \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_7(x))
Eq(F_7(x), F_29(x) + F_8(x))
Eq(F_8(x), F_13(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_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_5(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x)*F_9(x))
Eq(F_16(x), F_17(x) + F_24(x))
Eq(F_17(x), F_18(x) + F_9(x))
Eq(F_18(x), F_10(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_23(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_10(x) + F_19(x))
Eq(F_23(x), F_12(x)*F_18(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_20(x) + F_26(x) + F_27(x))
Eq(F_26(x), F_12(x)*F_24(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_10(x) + F_25(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_31(x)*F_9(x)**2)
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_9(x)**2)
Eq(F_33(x), F_34(x)*F_35(x))
Eq(F_34(x), F_10(x) + F_9(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_1(x) + F_36(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"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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_and_row_and_col_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": [1, 0, 2, 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], "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], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 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": [1, 0, 2, 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_placement", "direction": 3, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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": 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