0213_0231_1230_1302
Counting sequence:
1, 1, 2, 6, 20, 66, 212, 664, 2036, 6134, 18212, 53414, 155048, 446128, 1274024, 3614634, 10197308, 28624986, 80001500, 222719752, 617887388, 1708856606, 4712826452, 12964402766, 35581102160, 97447333216, 266367562832, 726811116114, 1979934791276, 5385461291394, 14628017732612, 39680896930744, 107510153700548, 290953969504838, 786570027045764, 2124307596735734, 5731791093723128, 15451833485982736, 41620583028245240, 112019785969133562, 301271546686015388, 809683367858588586, 2174611463830871852, 5836761505565622376, 15656672141475473900, 41973642802269267566, 112464423025971510452, 301179743072200851614, 806154998615751648416, 2156765730846046362304, 5767503470894615729312, 15416348105052237607074, 41189989977670667500748, 110008366086446016050898, 293690892486667517131604, 783776920855970600469784, 2090921916575286156860948, 5576122363373199050107286, 14865563739567401915601380, 39617791036909366792606022, 105551357385907484846767688, 281129703055530680843368816, 748554469714205738661658952, 1992590438241162861132608586, 5302670324114451394034271740, 14107724240416542147480367866, 37523960039278901987852904572, 99782225102148679826333790664, 265273465308432330619323504956, 705072351740662119479615943614, 1873595382661723745182706049812, 4977634993645291437004906364654, 13221421397875901763867720209648, 35111043401682033032948324427808, 93222839613257599482350462050544, 247466453655833944298590029794034, 656792325203747551160942145172268, 1742848955290897850025186636388386, 4623954036835354836655685228576420, 12265673210397793498255599793280632, 32530846263777276267174196080210788, 86263547534084718155868417813667046, 228712061528660791876624092177629252, 606292750669600947104885191097098646, 1606974266143672467564289758017814680, 4258634161135513346155492210465617616, 11284132481724045676117682237352119192, 29895371964050896810554990982699535514, 79191605186019957441961021823497442396, 209746700240787983259325665411732160074, 555460643701128388507822684204356785612, 1470804418710248734288020261922553266984, 3894048027807942816760975688468652868748, 10308456723001212656195063034546964610318, 27285577900680887940039587211595018484084, 72213927199210634528408949138717763260734, 191098898597975848830506208784124309281088, 505645203077627179914738413934973202609536, 1337781319182621565103590577647382567639360, 3538970145630494677054677782250016655544642, 9360998682641957690927751746373403367586188
Generating function in Maple syntax:
(x^5-3*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{x^{5}-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:
(x**5 - 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)-x^5+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)-x^{5}+3 x^{4}-15 x^{3}+17 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/25*((-5*n-4)*5^(1/2)+15*n)*(3/2-1/2*5^(1/2))^(-n)+1/25*((5*n+4)*5^(1/2)+15*n)*(3/2+1/2*5^(1/2))^(-n)+2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-5 n -4\right) \sqrt{5}+15 n \right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{25}+\\\frac{\left(\left(5 n +4\right) \sqrt{5}+15 n \right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{25}+2^{-1+n} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(n+5) = 2*a(n)-13*a(n+1)+28*a(n+2)-23*a(n+3)+8*a(n+4), n >= 6
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(5\right) = 66
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 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21331/
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[12,x]*F[4,x]
F[4,x] = F[40,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[26,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[25,x]
F[20,x] = 0
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[12,x]
F[24,x] = F[19,x]
F[25,x] = F[12,x]*F[14,x]
F[26,x] = F[20,x]+F[27,x]+F[39,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[10,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[10,x]*F[12,x]
F[32,x] = F[26,x]+F[33,x]
F[33,x] = 2*F[20,x]+F[34,x]+F[38,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[30,x]
F[37,x] = F[33,x]
F[38,x] = F[12,x]*F[26,x]
F[39,x] = F[12,x]*F[13,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[55,x]
F[44,x] = F[45,x]*F[52,x]
F[45,x] = F[46,x]+F[50,x]
F[46,x] = F[47,x]*F[5,x]
F[47,x] = F[1,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[47,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[47,x]*F[48,x]*F[8,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = 2*F[53,x]+F[1,x]
F[55,x] = F[40,x]*F[47,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{40}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{26}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)+F_{39}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{55}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right) F_{52}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right) F_{5}\! \left(x \right)
F_{47}\! \left(x \right) = F_{1}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right) F_{48}\! \left(x \right) F_{8}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{53}\! \left(x \right)+F_{1}\! \left(x \right)
F_{55}\! \left(x \right) = F_{40}\! \left(x \right) F_{47}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_40(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_26(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_25(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_12(x))
Eq(F_24(x), F_19(x))
Eq(F_25(x), F_12(x)*F_14(x))
Eq(F_26(x), F_20(x) + F_27(x) + F_39(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_10(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_10(x)*F_12(x))
Eq(F_32(x), F_26(x) + F_33(x))
Eq(F_33(x), 2*F_20(x) + F_34(x) + F_38(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_30(x))
Eq(F_37(x), F_33(x))
Eq(F_38(x), F_12(x)*F_26(x))
Eq(F_39(x), F_12(x)*F_13(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_55(x))
Eq(F_44(x), F_45(x)*F_52(x))
Eq(F_45(x), F_46(x) + F_50(x))
Eq(F_46(x), F_47(x)*F_5(x))
Eq(F_47(x), F_1(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_47(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_47(x)*F_48(x)*F_8(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_1(x) + 2*F_53(x))
Eq(F_55(x), F_40(x)*F_47(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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 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, 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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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