0132_0231_1302_2031_3102
Counting sequence:
1, 1, 2, 6, 19, 57, 165, 471, 1339, 3803, 10799, 30663, 87063, 247199, 701871, 1992815, 5658175, 16065183, 45613663, 129510271, 367716799, 1044053439, 2964367103, 8416688255, 23897391487, 67851547135, 192649999103, 546988590847, 1553057461247, 4409575479807, 12520049255935, 35548010027007, 100930994043903, 286572034579455, 813660182196223, 2310214578554879, 6559361654593535, 18623908668547071, 52878617212923903, 150137557486694399, 426283578432176127, 1210341318207774719, 3436506074075783167, 9757226180336386047, 27703563061457960959, 78658359672637865983, 223333638805392392191, 634108242653934026751, 1800415134739442270207, 5111894845952817758207, 14514135330161370005503, 41209792206313093529599, 117006417196512529612799, 332214770621120484343807, 943252998111058005983231, 2678165744352775145783295, 7604080526209109527887871, 21590165123647453540384767, 61300669878458926341554175, 174050180071664110260453375, 494178371020115642756366335, 1403114103550723982666760191, 3983843290265761580162482175, 11311273488900381675495620607, 32115949001639928346666074111, 91186388631489709845006778367, 258904305497199889344354058239, 735103565466120574690721267711, 2087169817138662161075482394623, 5926073617684566124227752624127, 16825822333115857001990523453439, 47773334496002487833202048106495, 135642195886915655573301603860479, 385127090229884384964379269922815, 1093486246363879893230714760462335, 3104721016309813344211945169879039, 8815193260243502573853680177512447, 25028861468462516705028369551458303, 71064114865495683213122647835541503, 201771400108619671180084473491357695, 572886863909420985277904041726181375, 1626590084934196622817523568610377727, 4618355684316341892518647054519500799, 13112836125347423049360302027088855039, 37231101933992717873001667027498172415, 105709774603213863079602754110676271103, 300140362927579426925124952386890104831, 852184557165447691128092127159851679743, 2419596326157900782971074207313198841855, 6869927801824971221222178225387168858111, 19505695016999177135260600326615583424511, 55382261513012625222095836819540055949311, 157246121587703734789785302422472227815423, 446466830332762792541279063064959854313471, 1267647358006261481450166716209030617890815, 3599214353869402225215121217421227509940223, 10219201644118088560142376130978707276824575, 29015243877007268302609353086647980303319039, 82382597639295523420298150257519547125858303, 233907818337389197591039940945443681843806207, 664131370546389233288185868924992499478429695
Generating function in Maple syntax:
(x^4-2*x^3+5*x^2-4*x+1)/(x-1)/(2*x^3-4*x^2+4*x-1)
Generating function in latex syntax:
\frac{x^{4}-2 x^{3}+5 x^{2}-4 x +1}{\left(x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(x**4 - 2*x**3 + 5*x**2 - 4*x + 1)/((x - 1)*(2*x**3 - 4*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x^3-4*x^2+4*x-1)*F(x)-x^4+2*x^3-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)-x^{4}+2 x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
-36/11*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-6*_Z^3+8*_Z^2-5*_Z+1))+76/11*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-6*_Z^3+8*_Z^2-5*_Z+1))-137/22*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-6*_Z^3+8*_Z^2-5*_Z+1))+35/22*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-6*_Z^3+8*_Z^2-5*_Z+1))+piecewise(n = 0,1/2,0)
Explicit closed form in latex syntax:
-\frac{36 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{11}+\frac{76 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{11}-\frac{137 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{22}+\frac{35 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{22}+\left(\left\{\begin{array}{cc}\frac{1}{2} & n =0 \\ 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(n+3) = 2*a(n)-4*a(n+1)+4*a(n+2)+1, n >= 5
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(n +3\right) = 2 a \! \left(n \right)-4 a \! \left(n +1\right)+4 a \! \left(n +2\right)+1, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19105/
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[25,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[11,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[21,x]*F[4,x]
F[24,x] = F[18,x]
F[25,x] = F[26,x]+F[38,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[1,x]+F[14,x]
F[30,x] = F[26,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[26,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[34,x]*F[4,x]
F[37,x] = F[31,x]
F[38,x] = F[39,x]+F[40,x]+F[60,x]
F[39,x] = 0
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[44,x]
F[42,x] = F[43,x]+F[7,x]
F[43,x] = F[19,x]
F[44,x] = F[38,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[59,x]
F[48,x] = F[49,x]+F[57,x]
F[49,x] = F[39,x]+F[50,x]+F[55,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[11,x]+F[18,x]
F[53,x] = F[49,x]+F[54,x]
F[54,x] = F[46,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[26,x]+F[49,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[48,x]
F[59,x] = F[54,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[56,x]+F[62,x]
F[62,x] = F[31,x]+F[54,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_{25}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{38}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{26}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{31}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{60}\! \left(x \right)
F_{39}\! \left(x \right) = 0
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{7}\! \left(x \right)
F_{43}\! \left(x \right) = F_{19}\! \left(x \right)
F_{44}\! \left(x \right) = F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{59}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{57}\! \left(x \right)
F_{49}\! \left(x \right) = F_{39}\! \left(x \right)+F_{50}\! \left(x \right)+F_{55}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{46}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{26}\! \left(x \right)+F_{49}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{56}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{31}\! \left(x \right)+F_{54}\! \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_25(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_11(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_21(x)*F_4(x))
Eq(F_24(x), F_18(x))
Eq(F_25(x), F_26(x) + F_38(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_1(x) + F_14(x))
Eq(F_30(x), F_26(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_26(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_34(x)*F_4(x))
Eq(F_37(x), F_31(x))
Eq(F_38(x), F_39(x) + F_40(x) + F_60(x))
Eq(F_39(x), 0)
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_44(x))
Eq(F_42(x), F_43(x) + F_7(x))
Eq(F_43(x), F_19(x))
Eq(F_44(x), F_38(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_59(x))
Eq(F_48(x), F_49(x) + F_57(x))
Eq(F_49(x), F_39(x) + F_50(x) + F_55(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_11(x) + F_18(x))
Eq(F_53(x), F_49(x) + F_54(x))
Eq(F_54(x), F_46(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_26(x) + F_49(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_48(x))
Eq(F_59(x), F_54(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_56(x) + F_62(x))
Eq(F_62(x), F_31(x) + F_54(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, 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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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, 3, 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, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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