0132_0231_1320_3021_3120
Counting sequence:
1, 1, 2, 6, 19, 56, 153, 395, 980, 2364, 5587, 13001, 29886, 68018, 153509, 343959, 765832, 1695608, 3735399, 8191829, 17891138, 38928174, 84410137, 182451971, 393215724, 845151956, 1811939003, 3875536545, 8271167110, 17616076394, 37446745677, 79456894511, 168309030416, 355945414128, 751619276239, 1584842931629, 3337189588362, 7017976560998, 14740327759169, 30923764530459, 64802466561268, 135652247076044, 283673999965347, 592636767370361, 1236950581247054, 2579454278761506, 5374412836568053, 11188630324247495, 23274462136761240, 48378511622142824, 100486566685702967, 208572957742594821, 432627039204276946, 896779275800148638, 1857734846290328169, 3846074081774402099, 7957860541563664892, 16456153038411790788, 34011184385901984139, 70256154186979735889, 145051936798348933398, 299327245633552644314, 617389465716966553757, 1272825341085959059551, 2622896422980576868384, 5402590170587684927456, 11123386676446859622303, 22892409395473553553245, 47094537620180485273370, 96845406386975145981654, 199077262043473481037457, 409074996578583016633931, 840286086045617495209476, 1725434653678496619951548, 3541774862152233910269299, 7267722017136383983875369, 14908510986419469939635422, 30572600610098083113464978, 62675247960645931276170309, 128448368334054349812528119, 263168039357359588468847528, 539029799820672783272112984, 1103749273308156436506735367, 2259482356859741927525839541, 4624141260025956593251119714, 9461053464304087921250529806, 19352484520390983828696461753, 39575395630904500663181373795, 80910987255167901404734943500, 165381052123281271099804872884, 337957630724908815047460909147, 690461056911420848324986531841, 1410323189755869478178827268006, 2880067501397437209552812502858, 5880215186605556305770840059629, 12003066620911047145421908471439, 24496357497379104879703870139952, 49983067026186513979327039663568, 101966645155858202482891064077679, 207973926599943886183052869627149, 424108353938856999138240766143658
Generating function in Maple syntax:
(x^7+2*x^6-13*x^5+34*x^4-42*x^3+26*x^2-8*x+1)/(x-1)^3/(2*x-1)^3
Generating function in latex syntax:
\frac{x^{7}+2 x^{6}-13 x^{5}+34 x^{4}-42 x^{3}+26 x^{2}-8 x +1}{\left(x -1\right)^{3} \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(x**7 + 2*x**6 - 13*x**5 + 34*x**4 - 42*x**3 + 26*x**2 - 8*x + 1)/((x - 1)**3*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
-(x-1)^3*(2*x-1)^3*F(x)+x^7+2*x^6-13*x^5+34*x^4-42*x^3+26*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{3} \left(2 x -1\right)^{3} F \! \left(x \right)+x^{7}+2 x^{6}-13 x^{5}+34 x^{4}-42 x^{3}+26 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/32*(n^2+7*n+6)*2^n-1/2*n^2+1/2*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(n^{2}+7 n +6\right) 2^{n}}{32}-\frac{n^{2}}{2}+\frac{n}{2} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 153
a(7) = 395
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)+1/2*(-1+n)*(n-6), n >= 8
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) = 56
a \! \left(6\right) = 153
a \! \left(7\right) = 395
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)+\frac{\left(-1+n \right) \left(n -6\right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21739/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[17,x]*F[7,x]
F[7,x] = F[56,x]+F[8,x]
F[8,x] = F[44,x]+F[9,x]
F[9,x] = F[10,x]+F[27,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[17,x]
F[13,x] = F[14,x]+F[18,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[22,x]
F[19,x] = F[20,x]
F[20,x] = F[17,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]+F[24,x]+F[26,x]
F[23,x] = 0
F[24,x] = F[17,x]*F[25,x]
F[25,x] = F[15,x]+F[22,x]
F[26,x] = F[17,x]*F[18,x]
F[27,x] = F[19,x]+F[28,x]
F[28,x] = F[23,x]+F[29,x]+F[34,x]
F[29,x] = F[17,x]*F[30,x]
F[30,x] = F[11,x]+F[31,x]
F[31,x] = F[23,x]+F[29,x]+F[32,x]
F[32,x] = F[17,x]*F[33,x]
F[33,x] = F[18,x]
F[34,x] = F[17,x]*F[35,x]
F[35,x] = F[18,x]+F[36,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[17,x]*F[39,x]
F[39,x] = F[19,x]+F[37,x]
F[40,x] = 2*F[23,x]+F[41,x]+F[43,x]
F[41,x] = F[17,x]*F[42,x]
F[42,x] = F[22,x]+F[40,x]
F[43,x] = F[17,x]*F[36,x]
F[44,x] = F[45,x]*F[53,x]
F[45,x] = F[46,x]
F[46,x] = F[17,x]*F[47,x]
F[47,x] = F[14,x]+F[48,x]
F[48,x] = F[19,x]+F[49,x]
F[49,x] = F[23,x]+F[50,x]+F[52,x]
F[50,x] = F[17,x]*F[51,x]
F[51,x] = F[15,x]+F[49,x]
F[52,x] = F[17,x]*F[48,x]
F[53,x] = F[10,x]+F[54,x]
F[54,x] = F[19,x]+F[55,x]
F[55,x] = F[23,x]+F[24,x]+F[34,x]
F[56,x] = F[57,x]
F[57,x] = F[10,x]*F[19,x]*F[58,x]
F[58,x] = F[21,x]+F[45,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_{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_{17}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{56}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{44}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{27}\! \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_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \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_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \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_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{17}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right) F_{18}\! \left(x \right)
F_{27}\! \left(x \right) = F_{19}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{29}\! \left(x \right) = F_{17}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{17}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)
F_{34}\! \left(x \right) = F_{17}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{17}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{17}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{22}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{17}\! \left(x \right) F_{36}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right) F_{53}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{17}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{19}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{23}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{17}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{15}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{17}\! \left(x \right) F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{19}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{34}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{10}\! \left(x \right) F_{19}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)+F_{45}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_17(x)*F_7(x))
Eq(F_7(x), F_56(x) + F_8(x))
Eq(F_8(x), F_44(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_27(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_17(x))
Eq(F_13(x), F_14(x) + F_18(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) + F_22(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_17(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_26(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_17(x)*F_25(x))
Eq(F_25(x), F_15(x) + F_22(x))
Eq(F_26(x), F_17(x)*F_18(x))
Eq(F_27(x), F_19(x) + F_28(x))
Eq(F_28(x), F_23(x) + F_29(x) + F_34(x))
Eq(F_29(x), F_17(x)*F_30(x))
Eq(F_30(x), F_11(x) + F_31(x))
Eq(F_31(x), F_23(x) + F_29(x) + F_32(x))
Eq(F_32(x), F_17(x)*F_33(x))
Eq(F_33(x), F_18(x))
Eq(F_34(x), F_17(x)*F_35(x))
Eq(F_35(x), F_18(x) + F_36(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_17(x)*F_39(x))
Eq(F_39(x), F_19(x) + F_37(x))
Eq(F_40(x), 2*F_23(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_17(x)*F_42(x))
Eq(F_42(x), F_22(x) + F_40(x))
Eq(F_43(x), F_17(x)*F_36(x))
Eq(F_44(x), F_45(x)*F_53(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_17(x)*F_47(x))
Eq(F_47(x), F_14(x) + F_48(x))
Eq(F_48(x), F_19(x) + F_49(x))
Eq(F_49(x), F_23(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_17(x)*F_51(x))
Eq(F_51(x), F_15(x) + F_49(x))
Eq(F_52(x), F_17(x)*F_48(x))
Eq(F_53(x), F_10(x) + F_54(x))
Eq(F_54(x), F_19(x) + F_55(x))
Eq(F_55(x), F_23(x) + F_24(x) + F_34(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_10(x)*F_19(x)*F_58(x))
Eq(F_58(x), F_21(x) + F_45(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:
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