0132_0213_0321_1302_2301_3120
Counting sequence:
1, 1, 2, 6, 18, 42, 86, 166, 314, 594, 1134, 2190, 4274, 8410, 16646, 33078, 65898, 131490, 262622, 524830, 1049186, 2097834, 4195062, 8389446, 16778138, 33555442, 67109966, 134218926, 268436754, 536872314, 1073743334, 2147485270, 4294969034, 8589936450, 17179871166, 34359740478, 68719478978, 137438955850, 274877909462, 549755816550, 1099511630586, 2199023258514, 4398046514222, 8796093025486, 17592186047858, 35184372092442, 70368744181446, 140737488359286, 281474976714794, 562949953425634, 1125899906847134, 2251799813689950, 4503599627375394, 9007199254746090, 18014398509487286, 36028797018969478, 72057594037933658, 144115188075861810, 288230376151717902, 576460752303429870, 1152921504606853586, 2305843009213700794, 4611686018427394982, 9223372036854783126, 18446744073709559178, 36893488147419111042, 73786976294838214526, 147573952589676421246, 295147905179352834434, 590295810358705660554, 1180591620717411312534, 2361183241434822616230, 4722366482869645223354, 9444732965739290437330, 18889465931478580865006, 37778931862957161720078, 75557863725914323429938, 151115727451828646849370, 302231454903657293687942, 604462909807314587364790, 1208925819614629174718186, 2417851639229258349424674, 4835703278458516698837342, 9671406556917033397662366, 19342813113834066795312098, 38685626227668133590611242, 77371252455336267181209206, 154742504910672534362404806, 309485009821345068724795674, 618970019642690137449577074, 1237940039285380274899139534, 2475880078570760549798264110, 4951760157141521099596512914, 9903520314283042199193010170, 19807040628566084398386004326, 39614081257132168796771992278, 79228162514264337593543967818, 158456325028528675187087918530, 316912650057057350374175819582, 633825300114114700748351621310, 1267650600228229401496703224386
Generating function in Maple syntax:
-(6*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{6 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:
(-6*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)+6*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)+6 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),1,10-10*n+2*n^2+2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 10-10 n +2 n^{2}+2^{n} & \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) = 42
a(n+1) = -2*n^2+2*a(n)+14*n-18, 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) = 18
a \! \left(5\right) = 42
a \! \left(n +1\right) = -2 n^{2}+2 a \! \left(n \right)+14 n -18, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22239/
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[15,x]*F[4,x]
F[4,x] = F[22,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[15,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[15,x]*F[16,x]
F[11,x] = F[12,x]+F[19,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]*F[16,x]
F[15,x] = x
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[15,x]*F[16,x]
F[19,x] = F[20,x]
F[20,x] = F[15,x]*F[16,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]
F[23,x] = F[15,x]*F[24,x]
F[24,x] = F[25,x]+F[47,x]
F[25,x] = F[26,x]+F[44,x]
F[26,x] = F[27,x]+F[41,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[16,x]^2*F[29,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[15,x]*F[29,x]
F[31,x] = F[32,x]+F[38,x]
F[32,x] = F[33,x]
F[33,x] = F[15,x]*F[16,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[13,x]
F[36,x] = F[17,x]*F[37,x]
F[37,x] = F[30,x]
F[38,x] = F[37,x]*F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[15,x]*F[16,x]*F[35,x]
F[41,x] = F[16,x]*F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[15,x]*F[37,x]
F[44,x] = F[16,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[15,x]*F[16,x]*F[21,x]*F[37,x]
F[47,x] = F[16,x]*F[17,x]*F[48,x]
F[48,x] = F[29,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_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \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_{15}\! \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_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{19}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{47}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{44}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{41}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16} \left(x \right)^{2} F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right) F_{29}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{38}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right)
F_{36}\! \left(x \right) = F_{17}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{30}\! \left(x \right)
F_{38}\! \left(x \right) = F_{37}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{35}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right) F_{37}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{21}\! \left(x \right) F_{37}\! \left(x \right)
F_{47}\! \left(x \right) = F_{16}\! \left(x \right) F_{17}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{29}\! \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_15(x)*F_4(x))
Eq(F_4(x), F_22(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_15(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_15(x)*F_16(x))
Eq(F_11(x), F_12(x) + F_19(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x)*F_16(x))
Eq(F_15(x), x)
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_15(x)*F_16(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_15(x)*F_16(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_15(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_47(x))
Eq(F_25(x), F_26(x) + F_44(x))
Eq(F_26(x), F_27(x) + F_41(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_16(x)**2*F_29(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_15(x)*F_29(x))
Eq(F_31(x), F_32(x) + F_38(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_15(x)*F_16(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_13(x))
Eq(F_36(x), F_17(x)*F_37(x))
Eq(F_37(x), F_30(x))
Eq(F_38(x), F_37(x)*F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_15(x)*F_16(x)*F_35(x))
Eq(F_41(x), F_16(x)*F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_15(x)*F_37(x))
Eq(F_44(x), F_16(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_15(x)*F_16(x)*F_21(x)*F_37(x))
Eq(F_47(x), F_16(x)*F_17(x)*F_48(x))
Eq(F_48(x), F_29(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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 1, 3], "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, 3, 0, 2], "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, 1, 3], "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, 3, 0, 2], "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], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], 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