0132_0213_0231_0312_0321_1032_1302_3120
Counting sequence:
1, 1, 2, 6, 16, 39, 94, 218, 498, 1122, 2498, 5506, 12034, 26114, 56322, 120834, 258050, 548866, 1163266, 2457602, 5177346, 10878978, 22806530, 47710210, 99614722, 207618050, 432013314, 897581058, 1862270978, 3858759682, 7985954818, 16508780546, 34091302914, 70330089474, 144955146242, 298500227074, 614180323330, 1262720385026, 2594160246786, 5325759447042, 10926396801026, 22402549415938, 45904610459650, 94008244174850, 192414534860802, 393625162743810, 804842511532034, 1644869395152898, 3360107534483458, 6860952557322242, 14003380091355138, 28569710136131586, 58265320179105794, 118782440171896834, 242068479971164162, 493144159197069314, 1004302716903620610, 2044634230826205186, 4161326055690338306, 8466767299456532482, 17221764975064776706, 35019990702432976898, 71192902909472800770, 144691648828159295490, 293994983674745978882, 597213339386346733570, 1212873422846403018754, 2462640333840225140738, 4999067643975288487938, 10145709240540253388802, 20586566386259859603458, 41763428582878424858626, 84707448786474261020674, 171776080814383344648194, 348274528111636334510082, 705993789189011959447554, 1430877044309502499749890, 2899533020481962161209346, 5874623904689838645837826, 11900363536831505938513922, 24102958528566669170704386, 48810379966940652928761858, 98829685753495935032229890, 200077223146221128413872130, 404990149570900773526568962, 819651705698718580450787330, 1658646224511271227696873474, 3355978075250210588984344578, 6789327402955757445149884418, 13733397310822187424662159362, 27776279631465719918049099778, 56171529282574129973547761666, 113580998604433640221994647554, 229637877287438040993787543554, 464227514732017603087171584002, 938358549778318248373536161794, 1896524140185202581145458311170, 3832662361627537331087688597506, 7744552885769338999768921145346, 15647562096567206674724930191362, 31612036843191470699824036184066
Generating function in Maple syntax:
-(4*x^8-4*x^7+3*x^6-x^5-2*x^4+5*x^2-4*x+1)/(x-1)/(-1+2*x)^2
Generating function in latex syntax:
-\frac{4 x^{8}-4 x^{7}+3 x^{6}-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:
(-4*x**8 + 4*x**7 - 3*x**6 + 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)+4*x^8-4*x^7+3*x^6-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)+4 x^{8}-4 x^{7}+3 x^{6}-x^{5}-2 x^{4}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,n = 4,16,n = 5,39,2+1/16*(4*n-1)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 16 & n =4 \\ 39 & n =5 \\ 2+\frac{\left(4 n -1\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) = 39
a(6) = 94
a(7) = 218
a(8) = 498
a(n+2) = -4*a(n)+4*a(n+1)+2, n >= 9
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) = 39
a \! \left(6\right) = 94
a \! \left(7\right) = 218
a \! \left(8\right) = 498
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+2, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/23282/
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[13,x]*F[4,x]
F[4,x] = F[18,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]*F[15,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[14,x]
F[13,x] = x
F[14,x] = 2*F[12,x]+F[1,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[17,x]
F[17,x] = 2*F[16,x]+F[1,x]
F[18,x] = F[19,x]
F[19,x] = F[13,x]*F[20,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[10,x]*F[22,x]
F[22,x] = F[15,x]+F[23,x]
F[23,x] = F[1,x]+F[13,x]
F[24,x] = F[25,x]+F[34,x]
F[25,x] = F[22,x]*F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[13,x]*F[28,x]
F[28,x] = F[29,x]+F[31,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[13,x]*F[29,x]
F[31,x] = F[32,x]
F[32,x] = F[13,x]*F[29,x]*F[33,x]
F[33,x] = F[13,x]+F[14,x]
F[34,x] = F[13,x]^2*F[35,x]
F[35,x] = F[1,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{18}\! \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_{13}\! \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_{15}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = 2 F_{12}\! \left(x \right)+F_{1}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{1}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{31}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13} \left(x \right)^{2} F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_18(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_13(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_15(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_14(x))
Eq(F_13(x), x)
Eq(F_14(x), F_1(x) + 2*F_12(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_1(x) + 2*F_16(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_13(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_10(x)*F_22(x))
Eq(F_22(x), F_15(x) + F_23(x))
Eq(F_23(x), F_1(x) + F_13(x))
Eq(F_24(x), F_25(x) + F_34(x))
Eq(F_25(x), F_22(x)*F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_13(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_31(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_13(x)*F_29(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_13(x)*F_29(x)*F_33(x))
Eq(F_33(x), F_13(x) + F_14(x))
Eq(F_34(x), F_13(x)**2*F_35(x))
Eq(F_35(x), F_1(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, 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, 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, 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, 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, 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, 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, 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, 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, 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"}}, {"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": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [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, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 1], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, 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