0123_0132_0231_1203_2103
Counting sequence:
1, 1, 2, 6, 19, 55, 156, 444, 1269, 3629, 10374, 29650, 84743, 242211, 692288, 1978696, 5655497, 16164505, 46201290, 132052254, 377430971, 1078770959, 3083336740, 8812774740, 25188620381, 71993965061, 205772723022, 588138373866, 1681013604399, 4804663092475, 13732659492936, 39250605738464, 112185848023729, 320648924010609, 916476848732434, 2619468681687350, 7486949816387683, 21399155464229351, 61162938955452588, 174815548581890796, 499656761887224517, 1428115986963694045, 4081832625496653590, 11666669748577272834, 33345606130973307991, 95308213243766692819, 272409368599897350480, 778598838184053212280, 2225386571457975595161, 6360586671277304309321, 18179791017757785246042, 51961370629822133448526, 148515680685901870599179, 424486635788195872213695, 1213265179344023449982516, 3467747324189420615230084, 9911494790384238009119789, 28328975497883708606539189, 80969709386144355325692894, 231427142099312436610119450, 661463681990445139129444223, 1890591564253934753316984491, 5403677571642960268083014488, 15444759116864016063281742096, 44144118707183912953338084737, 126172457704838552571449495585, 360625368667481681360458173090, 1030737285238518436589826592614, 2946047182167280006887147269811, 8420374545340480370574972381079, 24067064476427642904148512812220, 68788340636586765221937366510620, 196610426343007693146012853640341, 561950751959544917861399651984781, 1606164299125069111468420494449062, 4590729253031843640133749418023986, 13121195064616022113086253118888359, 37502921743860381646316229737059715, 107190628018247104688933026499212896, 306371615881560641813836601715515304, 875669531499527246625065514622169449, 2502833450122988966042925428124325369, 7153583690787494502695413424464747754, 20446330385502143072296053563323117950, 58439579978840978173485803722873270363, 167031660142054263882942621886010091631, 477408898214398945553028942347409617988, 1364527275250987837094222116005738268596, 3900083747638300785959551635001561820157, 11147196186162421994103346975118376993125, 31860849882528750924550769291054126663790, 91064491759564107239696956765057041355594, 260279989077602594360562551249238498036431, 743930717728097270754748160223949333403739, 2126298355631310826992480424769774796857960, 6077373321762566019062649578864727637917760, 17370312305539689689671341574546834446371921, 49647723385944008418453351889993449164863441, 141902827885317523238736324523837790279852082, 405585819219073383025552262413796211396900246, 1159242977768926013635716497756637961968455939
Generating function in Maple syntax:
(2*x-1)/(2*x^4+x^3-x^2+3*x-1)
Generating function in latex syntax:
\frac{2 x -1}{2 x^{4}+x^{3}-x^{2}+3 x -1}
Generating function in sympy syntax:
(2*x - 1)/(2*x**4 + x**3 - x**2 + 3*x - 1)
Implicit equation for the generating function in Maple syntax:
(2*x^4+x^3-x^2+3*x-1)*F(x)-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{4}+x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)-2 x +1 = 0
Explicit closed form in Maple syntax:
-5/571*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4+_Z^3-_Z^2+3*_Z-1))+28/571*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4+_Z^3-_Z^2+3*_Z-1))+583/2284*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4+_Z^3-_Z^2+3*_Z-1))+11/2284*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4+_Z^3-_Z^2+3*_Z-1))
Explicit closed form in latex syntax:
-\frac{5 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{571}+\frac{28 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{571}+\frac{583 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{2284}+\frac{11 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{2284}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = 2*a(n)+a(n+1)-a(n+2)+3*a(n+3), n >= 4
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(n +4\right) = 2 a \! \left(n \right)+a \! \left(n +1\right)-a \! \left(n +2\right)+3 a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16104/
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[44,x]
F[17,x] = 0
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[26,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[24,x]+F[25,x]
F[24,x] = F[7,x]
F[25,x] = F[21,x]
F[26,x] = F[27,x]+F[37,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[2,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[2,x]*F[4,x]
F[33,x] = F[27,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[31,x]
F[37,x] = 2*F[17,x]+F[38,x]+F[40,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[31,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[16,x]
F[43,x] = F[37,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[30,x]+F[46,x]
F[46,x] = F[16,x]+F[47,x]
F[47,x] = F[38,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_{44}\! \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_{26}\! \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_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{37}\! \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_{2}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{17}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{31}\! \left(x \right)
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_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{16}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{30}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{16}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{38}\! \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_44(x))
Eq(F_17(x), 0)
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_26(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_24(x) + F_25(x))
Eq(F_24(x), F_7(x))
Eq(F_25(x), F_21(x))
Eq(F_26(x), F_27(x) + F_37(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_2(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_2(x)*F_4(x))
Eq(F_33(x), F_27(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_31(x))
Eq(F_37(x), 2*F_17(x) + F_38(x) + F_40(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_31(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_16(x))
Eq(F_43(x), F_37(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_30(x) + F_46(x))
Eq(F_46(x), F_16(x) + F_47(x))
Eq(F_47(x), F_38(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": [1, 2, 0, 3], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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": 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