0213_0231_2031_2130
Counting sequence:
1, 1, 2, 6, 20, 67, 220, 705, 2208, 6778, 20454, 60837, 178744, 519715, 1497698, 4282998, 12166952, 34363399, 96560596, 270117669, 752618436, 2089547506, 5782889802, 15958433481, 43924528240, 120614289607, 330487832390, 903766933230, 2467013429708, 6722990571883, 18292976580364, 49703411054217, 134868922378344, 365511418454890, 989438928054318, 2675520236021613, 7227497090850472, 19505390688670699, 52593575800613546, 141691653897123174, 381425505336640496, 1026001040705452111, 2757882307989699652, 7408144328056758381, 19886742418861554060, 53352161910800983522, 143049792915726168210, 383337295456751778705, 1026702297305653213408, 2748450164577577829263, 7353949767296810309774, 19667623422542161593822, 52576292066622846628244, 140489143753926539688115, 375247441684563991313788, 1001894200043186743174353, 2674001916689452573465392, 7134170815022317928810074, 19027121583138884241955446, 50728967869710133951403829, 135206492849235126084236056, 360248869356525458638752691, 959568480720918660524098802, 2555183311205893292975749206, 6802133303748093622444535480, 18102945416498565025653505687, 48165737548888487422788928180, 128119642232352749733091246005, 340710279570033177353055316308, 905837092778044664712827358418, 2407761597316918024663952425434, 6398503598678437103117530674777, 16999956298977907063504969028944, 45156930700118393540366569597015, 119925324907887679758403401190358, 318426945943573938125735247680718, 845324729581133972073538166757212, 2243646990864142931033934404027899, 5953946270564829308322173255586796, 15797082155464412434738212003786009, 41905641172252633694741348792090616, 111145973956083214032678694487228746, 294742337504244190308928086271181502, 781485076387205640137620073603683581, 2071714948342003323748456939945093000, 5491261900864557680437039911519311227, 14552875094249134875184464001769589050, 38562174269659160475209383583369634630, 102167276038079164796214814076718401408, 270645727926893160746883077275933248223, 716854501666271350951462682577326487844, 1898465484760773908300052418839996360957, 5027100481759247874029129999336992668444, 13309963840258328167855739097107471281090, 35235616149097852431702141800675459135778, 93268232057542693959753019073602328620833, 246850297264975459903057000738591352746432, 653254964011221815803730189371066978176799, 1728550290348778954256198666827233887426846, 4573330689501583431236006035408409124449214, 12098610429975366753679689277176286045264676
Generating function in Maple syntax:
-(3*x^4-15*x^3+17*x^2-7*x+1)/(2*x-1)/(x^2-3*x+1)^2
Generating function in latex syntax:
-\frac{3 x^{4}-15 x^{3}+17 x^{2}-7 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(-3*x**4 + 15*x**3 - 17*x**2 + 7*x - 1)/((2*x - 1)*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)^2*F(x)+3*x^4-15*x^3+17*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+3 x^{4}-15 x^{3}+17 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
1/25*(5*n-8*5^(1/2))*(3/2-1/2*5^(1/2))^(-n)+1/25*(5*n+8*5^(1/2))*(3/2+1/2*5^(1/2))^(-n)+2^n
Explicit closed form in latex syntax:
\frac{\left(5 n -8 \sqrt{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{25}+\frac{\left(5 n +8 \sqrt{5}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{25}+2^{n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+5) = 2*a(n)-13*a(n+1)+28*a(n+2)-23*a(n+3)+8*a(n+4), 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) = 20
a \! \left(n +5\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+28 a \! \left(n +2\right)-23 a \! \left(n +3\right)+8 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21352/
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[8,x]
F[4,x] = F[11,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[6,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[4,x]*F[5,x]
F[15,x] = F[16,x]*F[31,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]^2*F[8,x]
F[18,x] = F[19,x]+F[22,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[19,x]*F[8,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[8,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[27,x]+F[28,x]+F[30,x]
F[27,x] = 0
F[28,x] = F[29,x]*F[8,x]
F[29,x] = F[20,x]+F[26,x]
F[30,x] = F[22,x]*F[8,x]
F[31,x] = F[25,x]+F[32,x]
F[32,x] = F[33,x]+F[39,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[8,x]
F[35,x] = F[25,x]+F[36,x]
F[36,x] = F[33,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[36,x]*F[8,x]
F[39,x] = F[27,x]+F[40,x]+F[48,x]
F[40,x] = F[41,x]*F[8,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[23,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[42,x]*F[8,x]
F[45,x] = F[39,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[45,x]*F[8,x]
F[48,x] = F[32,x]*F[8,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{11}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{31}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18} \left(x \right)^{2} F_{8}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right)+F_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{22}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right) F_{8}\! \left(x \right)
F_{39}\! \left(x \right) = F_{27}\! \left(x \right)+F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right) F_{8}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{23}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{42}\! \left(x \right) F_{8}\! \left(x \right)
F_{45}\! \left(x \right) = F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{45}\! \left(x \right) F_{8}\! \left(x \right)
F_{48}\! \left(x \right) = F_{32}\! \left(x \right) F_{8}\! \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_8(x))
Eq(F_4(x), F_11(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_6(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_4(x)*F_5(x))
Eq(F_15(x), F_16(x)*F_31(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)**2*F_8(x))
Eq(F_18(x), F_19(x) + F_22(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_19(x)*F_8(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_8(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_30(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_29(x)*F_8(x))
Eq(F_29(x), F_20(x) + F_26(x))
Eq(F_30(x), F_22(x)*F_8(x))
Eq(F_31(x), F_25(x) + F_32(x))
Eq(F_32(x), F_33(x) + F_39(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_8(x))
Eq(F_35(x), F_25(x) + F_36(x))
Eq(F_36(x), F_33(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_36(x)*F_8(x))
Eq(F_39(x), F_27(x) + F_40(x) + F_48(x))
Eq(F_40(x), F_41(x)*F_8(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_23(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_42(x)*F_8(x))
Eq(F_45(x), F_39(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_45(x)*F_8(x))
Eq(F_48(x), F_32(x)*F_8(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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, 3, 0], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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, 3, 0], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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, 3, 0], "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": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": 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