0123_0132_1230_2031_2103
Counting sequence:
1, 1, 2, 6, 19, 50, 122, 296, 714, 1709, 4063, 9610, 22643, 53195, 124687, 291745, 681688, 1591102, 3710579, 8647598, 20142860, 46899366, 109161878, 254017189, 590971909, 1374677288, 3197271455, 7435576711, 17290809805, 40205803975, 93484715860, 217357898090, 505355218541, 1174917622808, 2731553077608, 6350460153446, 14763708671308, 34322709223891, 79792918612829, 185500261374968, 431243570005669, 1002533983205713, 2330635077502069, 5418117633830095, 12595684411268618, 29281577529475436, 68071710364853273, 158248081579416132, 367883176121517662, 855226500251747044, 1988163850488833932, 4621926546746410787, 10744687304253581999, 24978388974556126222, 58067748851109149197, 134991211067209939213, 313816625926709699143, 729535391097503107685, 1695964475718467819210, 3942639902152701689792, 9165527307197119981743, 21307268728560634531574, 49533395401340617659238, 115151183278632613824340, 267694041049768613099078, 622313176433807763986993, 1446702675433471496607355, 3363175797163938299059510, 7818435369133429743762559, 18175657468214957746720303, 42253277978007146468563691, 98226955312973971765637453, 228349969191545866249942388, 530849278127834300357045874, 1234074857201194943259881759, 2868875995368832683051854410, 6669327569750260352322125464, 15504305612349808914111179162, 36043137760300925477795409170, 83790129749890421883858172561, 194788419569374911573509954449, 452828137387920694077450449452, 1052697703546071249505818387931, 2447225256850147406082949593251, 5689108503455064993645018125225, 13225572705009384701288843398811, 30745726373793829520150554406736, 71475142231050700012354330908902, 166159546680430812839391044362345, 386274082007900429013281789954860, 897978294994895825375260029739076, 2087546267835095443081457052689330, 4852956295865094144651491569533352, 11281754647548469886455871230658887, 26226897619906276375722346383305833, 60970139862613124611916456314526692, 141738378999485874159635524839425873, 329501754900330333495256912538428805, 765998646577861784944761694005052289, 1780730808956743237223088213229513371, 4139696888659591242525486049688642246
Generating function in Maple syntax:
(2*x^9+2*x^8-4*x^6-3*x^5+4*x^4+x^3+2*x^2-3*x+1)/(x^3-2*x^2+3*x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
\frac{2 x^{9}+2 x^{8}-4 x^{6}-3 x^{5}+4 x^{4}+x^{3}+2 x^{2}-3 x +1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(2*x**9 + 2*x**8 - 4*x**6 - 3*x**5 + 4*x**4 + x**3 + 2*x**2 - 3*x + 1)/((x**3 - 2*x**2 + 3*x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^3-2*x^2+3*x-1)*(x^3+x^2+x-1)*F(x)+2*x^9+2*x^8-4*x^6-3*x^5+4*x^4+x^3+2*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+2 x^{9}+2 x^{8}-4 x^{6}-3 x^{5}+4 x^{4}+x^{3}+2 x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
-7079/5566*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-1330/2783*Sum(_alpha^(3-n),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-8743/2783*Sum(_alpha^(2-n),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-4087/5566*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-20763/5566*Sum(_alpha^(-n),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+13561/5566*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+2*piecewise(n = 0,-5,n = 2,2,n = 3,1,0)
Explicit closed form in latex syntax:
-\frac{7079 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{5566}-\frac{1330 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{3-n}\right)}{2783}-\frac{8743 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{2783}-\frac{4087 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{5566}-\frac{20763 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{5566}+\frac{13561 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{5566}+2 \left(\left\{\begin{array}{cc}-5 & n =0 \\ 2 & n =2 \\ 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) = 50
a(6) = 122
a(7) = 296
a(8) = 714
a(9) = 1709
a(n+6) = -a(n)+a(n+1)-2*a(n+2)+a(n+3)-4*a(n+4)+4*a(n+5), n >= 10
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) = 50
a \! \left(6\right) = 122
a \! \left(7\right) = 296
a \! \left(8\right) = 714
a \! \left(9\right) = 1709
a \! \left(n +6\right) = -a \! \left(n \right)+a \! \left(n +1\right)-2 a \! \left(n +2\right)+a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right), \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16517/
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[19,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[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[32,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[27,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[16,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[2,x]
F[31,x] = F[28,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[40,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]+F[4,x]
F[39,x] = F[25,x]
F[40,x] = F[20,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[39,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_{19}\! \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_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \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_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{2}\! \left(x \right)
F_{31}\! \left(x \right) = F_{28}\! \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_{40}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
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_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{25}\! \left(x \right)
F_{40}\! \left(x \right) = F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \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_19(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) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_32(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_27(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_16(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_31(x))
Eq(F_30(x), F_2(x))
Eq(F_31(x), F_28(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_40(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x) + F_4(x))
Eq(F_39(x), F_25(x))
Eq(F_40(x), F_20(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_39(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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 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]]}], "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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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