0123_0132_0231_2103_2130
Counting sequence:
1, 1, 2, 6, 19, 51, 121, 278, 627, 1398, 3100, 6856, 15142, 33419, 73732, 162647, 358757, 791293, 1745283, 3849376, 8490101, 18725544, 41300526, 91091218, 200908048, 443116693, 977324678, 2155557481, 4754231735, 10485788231, 23127134029, 51008499882, 112502788087, 248132710298, 547273920576, 1207050629340, 2662233969082, 5871741858847, 12950534347144, 28563302663483, 62998347185929, 138947228719121, 306457760101847, 675913867389748, 1490774963498745, 3288007687099468, 7251929241588818, 15994633446676518, 35277274580452644, 77806478402494249, 171607590251665162, 378492455083783117, 834791388570060635, 1841190367391786587, 4060873189867356449, 8956537768304773694, 19754265904001334139, 43569404997870024894, 96095347764044823652, 211944961432090981616, 467459327862051988302, 1031014003488148800435, 2273972968408388582668, 5015405264678829153823, 11061824532845807108269, 24397622034100002799397, 53810649332878834752811, 118683123198603476614088, 261763868431306956027773, 577338386195492746808560, 1273359895589588970231414, 2808483659610484896490810, 6194305705416462539790392, 13661971306422514049812413, 30132426272455512996115854, 66459158250327488532022321, 146580287807077491113857279, 323293001886610495223830639, 713045162023548478979683829, 1572670611854174449073225170, 3468634225594959393370281215, 7650313613213467265720246498, 16873297838281108980513718408, 37215229902157177354397718276, 82080773417527821974515683298, 181034844673336752929545085255, 399284919248830683213487889040, 880650611915189188401491461635, 1942336068503715129732528008785, 4283957056256260942678543906873, 9448564724427711073758579275647, 20839465517359137277249686560348, 45962888090974535497177917027841, 101374340906376782068114413331604, 223588147330112701413478513223834, 493139182751199938324134943475790, 1087652706408776658716384300283468, 2398893560147666018846247113791057, 5290926303046531976016629171058194, 11669505312501840610749642642400149, 25737904185151347240345532398591651
Generating function in Maple syntax:
(x^8-2*x^7+3*x^6-4*x^4-3*x^2+3*x-1)/(x^3+2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{8}-2 x^{7}+3 x^{6}-4 x^{4}-3 x^{2}+3 x -1}{\left(x^{3}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**8 - 2*x**7 + 3*x**6 - 4*x**4 - 3*x**2 + 3*x - 1)/((x - 1)**2*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^3+2*x-1)*(x-1)^2*F(x)+x^8-2*x^7+3*x^6-4*x^4-3*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{3}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}-2 x^{7}+3 x^{6}-4 x^{4}-3 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
-7/4-3/2*n+263/236*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^3+2*_Z-1))-319/236*Sum(_alpha^(-n),_alpha = RootOf(_Z^3+2*_Z-1))+213/236*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^3+2*_Z-1))+piecewise(n = 0,5,n = 3,1,0)
Explicit closed form in latex syntax:
-\frac{7}{4}-\frac{3 n}{2}+\frac{263 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{236}-\frac{319 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{236}+\frac{213 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{236}+\left(\left\{\begin{array}{cc}5 & n =0 \\ 1 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\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) = 121
a(7) = 278
a(8) = 627
a(n) = -2*a(n+2)+a(n+3)-3*n-5, 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) = 19
a \! \left(5\right) = 51
a \! \left(6\right) = 121
a \! \left(7\right) = 278
a \! \left(8\right) = 627
a \! \left(n \right) = -2 a \! \left(n +2\right)+a \! \left(n +3\right)-3 n -5, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16167/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[15,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[4,x]
F[15,x] = F[16,x]+F[2,x]
F[16,x] = F[17,x]+F[18,x]+F[25,x]
F[17,x] = 0
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]+F[7,x]
F[21,x] = F[13,x]+F[17,x]+F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[21,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[35,x]
F[27,x] = F[2,x]+F[28,x]
F[28,x] = F[17,x]+F[29,x]+F[34,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]+F[4,x]
F[32,x] = F[13,x]+F[17,x]+F[33,x]
F[33,x] = F[4,x]*F[7,x]
F[34,x] = F[2,x]*F[4,x]
F[35,x] = F[16,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[12,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{15}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{2}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{17}\! \left(x \right) = 0
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{7}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{35}\! \left(x \right)
F_{27}\! \left(x \right) = F_{2}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right)+F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_15(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_4(x))
Eq(F_15(x), F_16(x) + F_2(x))
Eq(F_16(x), F_17(x) + F_18(x) + F_25(x))
Eq(F_17(x), 0)
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x) + F_7(x))
Eq(F_21(x), F_13(x) + F_17(x) + F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_21(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_35(x))
Eq(F_27(x), F_2(x) + F_28(x))
Eq(F_28(x), F_17(x) + F_29(x) + F_34(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x) + F_4(x))
Eq(F_32(x), F_13(x) + F_17(x) + F_33(x))
Eq(F_33(x), F_4(x)*F_7(x))
Eq(F_34(x), F_2(x)*F_4(x))
Eq(F_35(x), F_16(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_12(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"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]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [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": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [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, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 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