0213_0231_0321_1203_2130
Counting sequence:
1, 1, 2, 6, 19, 55, 148, 383, 968, 2406, 5905, 14351, 34609, 82950, 197823, 469858, 1112238, 2625541, 6183445, 14534371, 34107524, 79928795, 187088748, 437482258, 1022130233, 2386379251, 5568088185, 12985093566, 30268398139, 70528924590, 164286729298, 382574053333, 890684258953, 2073204665411, 4824846063500, 11226866182311, 26120217212704, 60763977591710, 141342789041473, 328749478172583, 764584532064273, 1778112824964646, 4134949681089015, 9615269509704106, 22358145160503094, 51987092480398405, 120876749304309773, 281047193461468883, 653441144953852068, 1519237738638261331, 3532142002259214980, 8211919440036306826, 19091767585391416809, 44385716940686181675, 103189757215962678073, 239898049601208407358, 557717239810988686963, 1296579154443717202966, 3014268587304793199770, 7007499800826754995221, 16290811598292038018849, 37872244212482681725395, 88043690105601293052748, 204679555217951691564383, 475827853110777445989208, 1106176785913314799243926, 2571571500558910300022385, 5978223370606016952977791, 13897773071903821468951089, 32308582325638893622557190, 75108700904876160526021423, 174607264537576727088449458, 405914080933261176408959678, 943638628238795080722195461, 2193699414606017306729504325, 5099743922710878315737857411, 11855489276033706175646310404, 27560714815034000724531823435, 64070980651242711182718912348, 148947143270688324364095811458, 346260466666602523630094254873, 804958660793651577702235697571, 1871303325686393898925465681465, 4350255388874392842431722478206, 10113122709515865862005876917419, 23510169743428852909798461773630, 54654537334016514067252974556898, 127056431492482290683043118226133, 295370462088391873702172349708793, 686653205685325907563184751972131, 1596275414513872973479254604207180, 3710890874543752394035570517350551, 8626776160857623972755394813538384, 20054824469136813763762313445979150, 46621816602514093059987752099927073, 108382586314676570265546085964792983, 251958968777238927721129190627367313, 585733587443078914253691399026169766, 1361665485365837814707178395736021095, 3165488398541399257580914567677997946, 7358868109441210486073344286377935174
Generating function in Maple syntax:
-(x^6-2*x^4+x^3-5*x^2+4*x-1)/(2*x-1)/(x^3-2*x^2+3*x-1)
Generating function in latex syntax:
-\frac{x^{6}-2 x^{4}+x^{3}-5 x^{2}+4 x -1}{\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(-x**6 + 2*x**4 - x**3 + 5*x**2 - 4*x + 1)/((2*x - 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^3-2*x^2+3*x-1)*F(x)+x^6-2*x^4+x^3-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+x^{6}-2 x^{4}+x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
-629/46*Sum(_alpha^(2-n),_alpha = RootOf(2*_Z^4-5*_Z^3+8*_Z^2-5*_Z+1))+662/23*Sum(_alpha^(1-n),_alpha = RootOf(2*_Z^4-5*_Z^3+8*_Z^2-5*_Z+1))-1921/46*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-5*_Z^3+8*_Z^2-5*_Z+1))+665/46*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-5*_Z^3+8*_Z^2-5*_Z+1))-1/2*piecewise(n = 0,1/4,n = 1,5/2,n = 2,1,0)
Explicit closed form in latex syntax:
-\frac{629 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{46}+\frac{662 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{23}-\frac{1921 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{46}+\frac{665 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{46}-\frac{\left(\left\{\begin{array}{cc}\frac{1}{4} & n =0 \\ \frac{5}{2} & n =1 \\ 1 & n =2 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)}{2}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 148
a(n+4) = -2*a(n)+5*a(n+1)-8*a(n+2)+5*a(n+3), n >= 7
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) = 55
a \! \left(6\right) = 148
a \! \left(n +4\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20247/
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[10,x]*F[4,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]
F[13,x] = F[9,x]
F[14,x] = F[15,x]+F[28,x]
F[15,x] = F[16,x]
F[16,x] = F[10,x]*F[17,x]
F[17,x] = F[18,x]+F[5,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[21,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]+F[25,x]
F[23,x] = F[24,x]
F[24,x] = F[10,x]*F[15,x]
F[25,x] = F[26,x]
F[26,x] = F[10,x]*F[27,x]
F[27,x] = F[22,x]
F[28,x] = F[29,x]+F[30,x]+F[40,x]
F[29,x] = 0
F[30,x] = F[10,x]*F[31,x]
F[31,x] = F[32,x]+F[34,x]
F[32,x] = F[33,x]+F[6,x]
F[33,x] = F[12,x]
F[34,x] = F[28,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[10,x]*F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[23,x]+F[39,x]
F[39,x] = F[36,x]
F[40,x] = F[10,x]*F[41,x]
F[41,x] = F[14,x]+F[42,x]
F[42,x] = F[43,x]+F[52,x]
F[43,x] = F[24,x]+F[29,x]+F[44,x]
F[44,x] = F[10,x]*F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[10,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = x^2
F[49,x] = F[23,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[10,x]*F[23,x]
F[52,x] = 2*F[29,x]+F[53,x]+F[57,x]
F[53,x] = F[10,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[11,x]
F[56,x] = F[39,x]
F[57,x] = F[10,x]*F[58,x]
F[58,x] = F[42,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \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) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{9}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{5}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{25}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{40}\! \left(x \right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x \right) = F_{10}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right)
F_{34}\! \left(x \right) = F_{28}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{23}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)
F_{40}\! \left(x \right) = F_{10}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{52}\! \left(x \right)
F_{43}\! \left(x \right) = F_{24}\! \left(x \right)+F_{29}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = x^{2}
F_{49}\! \left(x \right) = F_{23}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{10}\! \left(x \right) F_{23}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{53}\! \left(x \right)+F_{57}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right)
F_{56}\! \left(x \right) = F_{39}\! \left(x \right)
F_{57}\! \left(x \right) = F_{10}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{42}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_14(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_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_13(x))
Eq(F_13(x), F_9(x))
Eq(F_14(x), F_15(x) + F_28(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_10(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_5(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_21(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_25(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_10(x)*F_15(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_10(x)*F_27(x))
Eq(F_27(x), F_22(x))
Eq(F_28(x), F_29(x) + F_30(x) + F_40(x))
Eq(F_29(x), 0)
Eq(F_30(x), F_10(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_34(x))
Eq(F_32(x), F_33(x) + F_6(x))
Eq(F_33(x), F_12(x))
Eq(F_34(x), F_28(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_10(x)*F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_23(x) + F_39(x))
Eq(F_39(x), F_36(x))
Eq(F_40(x), F_10(x)*F_41(x))
Eq(F_41(x), F_14(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_52(x))
Eq(F_43(x), F_24(x) + F_29(x) + F_44(x))
Eq(F_44(x), F_10(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_10(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), x**2)
Eq(F_49(x), F_23(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_10(x)*F_23(x))
Eq(F_52(x), 2*F_29(x) + F_53(x) + F_57(x))
Eq(F_53(x), F_10(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_11(x))
Eq(F_56(x), F_39(x))
Eq(F_57(x), F_10(x)*F_58(x))
Eq(F_58(x), F_42(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, 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": [0, 3, 2, 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, 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": [0, 3, 2, 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, 3, 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