0132_0231_0321_1302_3102
Counting sequence:
1, 1, 2, 6, 19, 56, 162, 469, 1358, 3930, 11371, 32900, 95190, 275413, 796850, 2305518, 6670531, 19299776, 55839834, 161560789, 467442086, 1352445138, 3913014907, 11321483756, 32756326638, 94773525973, 274207218794, 793360783686, 2295422184211, 6641320206536, 19215260002770, 55595304170197, 160853292920894, 465395094598026, 1346522599213579, 3895879288884500, 11271905456635974, 32612882279450581, 94358499959484770, 273006428512275678, 789886550136243427, 2285370221815424144, 6612236972334580362, 19131113795460803797, 55351844857678679318, 160148894722244394882, 463357066899976112923, 1340625995693041302524, 3878818709623636584414, 11222544266977820676565, 32470066082695908050330, 93945291400357176643254, 271810896652331263232563, 786427530722028429155096, 2275362278314468377519234, 6583281097526039229867733, 19047336075707706170394158, 55109451686226016036353210, 159447581178022420398400939, 461327963999256657499637924, 1334755214192199584173219254, 3861832840933386935816821717, 11173399236606474538276521746, 32327875297270435847359182030, 93533892337071606875434667779, 270620600187157333855497864800, 782983658818941266385777290682, 2265398160943809679052703342805, 6554452024386802438483328467334, 18963925229854480169168836640562, 54868119987064257427128588139963, 158749338780111896989183584713996, 459307746812980393878759839283150, 1328910141632957799501353109415381, 3844921355646116798603666658390986, 11124469418931353384067166544323878, 32186307184413858952600539193411411, 93124294845602457070941784352692328, 269435516190256219028359146953313330, 779554868093993917919336734894070485, 2255477677782930448871534626434282782, 6525749197635309866751985438315404522, 18880879651310986987223095158300071755, 54627845111863511992660369289453376308, 158054154079550168724261983586690866214, 457296376429408968154571771787219365845, 1323090665432403759724774070353873983938, 3828083928026902648468274406447784318398, 11075753871507122121989192692563917702947, 32045359017359271236948851811951716758320, 92716491036624496885793911556186801208810, 268255621835528463664411268241736522274005, 776141092504679436581337596792926199129846, 2245600637751758839851189345249415677533154, 6497172064421654010301052975438793878017627, 18798197740522562064214644774652818354779420, 54388622432649549862045260039018492541386942, 157362013686269395542523241293280247025416661, 455293814107203760893953753390127885244421882, 1317296673500640986863428538955140393790622870, 3811320233767257990781378346061853790178280051
Generating function in Maple syntax:
(2*x-1)*(x^3+x-1)/(x-1)/(2*x^3-x^2+3*x-1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x^{3}+x -1\right)}{\left(x -1\right) \left(2 x^{3}-x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(2*x - 1)*(x**3 + x - 1)/((x - 1)*(2*x**3 - x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x^3-x^2+3*x-1)*F(x)-(2*x-1)*(x^3+x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x^{3}+x -1\right) = 0
Explicit closed form in Maple syntax:
-886/1899*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-3*_Z^3+4*_Z^2-4*_Z+1))+1025/1899*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-3*_Z^3+4*_Z^2-4*_Z+1))-442/633*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-3*_Z^3+4*_Z^2-4*_Z+1))+554/1899*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-3*_Z^3+4*_Z^2-4*_Z+1))+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
-\frac{886 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{1899}+\frac{1025 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{1899}-\frac{442 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{633}+\frac{554 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{1899}+\left(\left\{\begin{array}{cc}1 & 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)-a(n+1)+3*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)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+1, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18691/
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[17,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[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]+F[14,x]
F[17,x] = F[18,x]+F[34,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[26,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[11,x]
F[26,x] = F[18,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[4,x]
F[33,x] = F[31,x]
F[34,x] = F[35,x]+F[36,x]+F[67,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[45,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[35,x]+F[40,x]+F[43,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[16,x]+F[42,x]
F[42,x] = F[14,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[13,x]
F[45,x] = F[34,x]+F[46,x]
F[46,x] = 2*F[35,x]+F[47,x]+F[65,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[64,x]
F[49,x] = F[50,x]+F[60,x]
F[50,x] = F[35,x]+F[51,x]+F[62,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[11,x]+F[54,x]
F[54,x] = F[35,x]+F[40,x]+F[55,x]
F[55,x] = F[13,x]*F[4,x]
F[56,x] = F[50,x]+F[57,x]
F[57,x] = 2*F[35,x]+F[47,x]+F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[31,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[49,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[18,x]+F[50,x]
F[64,x] = F[60,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[59,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[59,x]+F[63,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_{17}\! \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_{11}\! \left(x \right)+F_{14}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \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_{26}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{67}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{14}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{13}\! \left(x \right)
F_{45}\! \left(x \right) = F_{34}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{47}\! \left(x \right)+F_{65}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{64}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{60}\! \left(x \right)
F_{50}\! \left(x \right) = F_{35}\! \left(x \right)+F_{51}\! \left(x \right)+F_{62}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{50}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{47}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{31}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{18}\! \left(x \right)+F_{50}\! \left(x \right)
F_{64}\! \left(x \right) = F_{60}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{59}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{59}\! \left(x \right)+F_{63}\! \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_17(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_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_34(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_26(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_11(x))
Eq(F_26(x), F_18(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_4(x))
Eq(F_33(x), F_31(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_67(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_45(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_35(x) + F_40(x) + F_43(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_16(x) + F_42(x))
Eq(F_42(x), F_14(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_13(x))
Eq(F_45(x), F_34(x) + F_46(x))
Eq(F_46(x), 2*F_35(x) + F_47(x) + F_65(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_64(x))
Eq(F_49(x), F_50(x) + F_60(x))
Eq(F_50(x), F_35(x) + F_51(x) + F_62(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_11(x) + F_54(x))
Eq(F_54(x), F_35(x) + F_40(x) + F_55(x))
Eq(F_55(x), F_13(x)*F_4(x))
Eq(F_56(x), F_50(x) + F_57(x))
Eq(F_57(x), 2*F_35(x) + F_47(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_31(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_49(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_18(x) + F_50(x))
Eq(F_64(x), F_60(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_59(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_59(x) + F_63(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:
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