0231_0312_0321_1320_2103
Counting sequence:
1, 1, 2, 6, 19, 55, 147, 376, 944, 2353, 5846, 14493, 35869, 88659, 218948, 540392, 1333271, 3288722, 8110929, 20001893, 49322287, 121617638, 299873158, 739384833, 1823048348, 4494924123, 11082666565, 27325283425, 67372723294, 166112718532, 409563500769, 1009809215920, 2489758543889, 6138680262343, 15135358670225, 37317313357470, 92008507914940, 226853552288859, 559323628108076, 1379052304593533, 3400151783015963, 8383316571596753, 20669664411579768, 50962530342051506, 125651749356257725, 309803339485341266, 763842203053219831, 1883307364257763635, 4643428462490785523, 11448703641506607216, 28227594352288222300, 69597144598707263641, 171596717573465248810, 423083930382361881985, 1043143567498973096961, 2571944770870381284547, 6341313037240377881684, 15634958997279059537428, 38549105115804660319823, 95045564588996180104638, 234341609769529468024045, 577785931481733525992217, 1424572370848188748214639, 3512384655295771784481446, 8660034561392353436393162, 21351932081643226546885097, 52644709485511569299658600, 129799281218028425193103847, 320029373689519869275132809, 789055217124527614814058377, 1945471843704686279600306710, 4796699410264812099076829664, 11826604073909635180808882225, 29159334775428919851633718980, 71894416962962555922763294045, 177260806196435526883334986483, 437049144297217700505490731433, 1077575797095583445312667732766, 2656839885485556082343378189448, 6550627989355967037235892101675, 16151039921282849129044082427272, 39821539394808247465815651351985, 98182841940886674274417918929047, 242076790553354053989182363173449, 596857570693407662760211558392328, 1471594855829520526529769617712390, 3628321941511106152675273957919261, 8945886198977045287334170051026254, 22056719655841212484940613089891051, 54382413453012942801509705433170063, 134083714129776846003121541713528275, 330592948221566309659935035690139800, 815100462596419840588620382501856392, 2009688251667178357824305365438943633, 4955029538351313166654017877067972590, 12216978282858620414340633438075434229, 30121830194679450921965681293522458341, 74267517979478641155602160054653026323, 183111855793093171319483003335206316084, 451475323859017832369179674657844153888, 1113144570409041396513352449099000960055
Generating function in Maple syntax:
-(x-1)*(x^6+2*x^5+x^4+2*x^3+2*x^2-3*x+1)/(x^2+x-1)/(3*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
-\frac{\left(x -1\right) \left(x^{6}+2 x^{5}+x^{4}+2 x^{3}+2 x^{2}-3 x +1\right)}{\left(x^{2}+x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(1 - x)*(x**6 + 2*x**5 + x**4 + 2*x**3 + 2*x**2 - 3*x + 1)/((x**2 + x - 1)*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(3*x^3-5*x^2+4*x-1)*F(x)+(x-1)*(x^6+2*x^5+x^4+2*x^3+2*x^2-3*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{6}+2 x^{5}+x^{4}+2 x^{3}+2 x^{2}-3 x +1\right) = 0
Explicit closed form in Maple syntax:
-707/155*Sum(_alpha^(-n+3),_alpha = RootOf(3*_Z^5-2*_Z^4-4*_Z^3+8*_Z^2-5*_Z+1))+67/465*Sum(_alpha^(2-n),_alpha = RootOf(3*_Z^5-2*_Z^4-4*_Z^3+8*_Z^2-5*_Z+1))+8896/1395*Sum(_alpha^(1-n),_alpha = RootOf(3*_Z^5-2*_Z^4-4*_Z^3+8*_Z^2-5*_Z+1))-10781/1395*Sum(_alpha^(-n),_alpha = RootOf(3*_Z^5-2*_Z^4-4*_Z^3+8*_Z^2-5*_Z+1))+3473/1395*Sum(_alpha^(-n-1),_alpha = RootOf(3*_Z^5-2*_Z^4-4*_Z^3+8*_Z^2-5*_Z+1))-1/3*piecewise(n = 0,13/9,n = 1,5/3,n = 2,1,0)
Explicit closed form in latex syntax:
-\frac{707 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{155}+\frac{67 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{465}+\frac{8896 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{1395}-\frac{10781 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{1395}+\frac{3473 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{1395}-\frac{\left(\left\{\begin{array}{cc}\frac{13}{9} & n =0 \\ \frac{5}{3} & n =1 \\ 1 & n =2 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)}{3}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 147
a(7) = 376
a(n+5) = -3*a(n)+2*a(n+1)+4*a(n+2)-8*a(n+3)+5*a(n+4), n >= 8
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) = 147
a \! \left(7\right) = 376
a \! \left(n +5\right) = -3 a \! \left(n \right)+2 a \! \left(n +1\right)+4 a \! \left(n +2\right)-8 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20626/
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[13,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[6,x]
F[10,x] = F[11,x]+F[4,x]
F[11,x] = F[12,x]
F[12,x] = F[4,x]*F[7,x]
F[13,x] = F[14,x]+F[2,x]
F[14,x] = F[15,x]+F[16,x]+F[41,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[24,x]
F[18,x] = F[19,x]+F[7,x]
F[19,x] = F[15,x]+F[20,x]+F[22,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[18,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[10,x]
F[24,x] = F[25,x]+F[29,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]
F[28,x] = F[2,x]+F[25,x]
F[29,x] = 2*F[15,x]+F[30,x]+F[32,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[24,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]+F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[37,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[13,x]+F[43,x]
F[43,x] = F[34,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[11,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{13}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{2}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{41}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \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_{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_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{37}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{34}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_13(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_6(x))
Eq(F_10(x), F_11(x) + F_4(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_4(x)*F_7(x))
Eq(F_13(x), F_14(x) + F_2(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_41(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_24(x))
Eq(F_18(x), F_19(x) + F_7(x))
Eq(F_19(x), F_15(x) + F_20(x) + F_22(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_18(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_10(x))
Eq(F_24(x), F_25(x) + F_29(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_2(x) + F_25(x))
Eq(F_29(x), 2*F_15(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_24(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(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_37(x))
Eq(F_37(x), F_38(x) + F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_37(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_13(x) + F_43(x))
Eq(F_43(x), F_34(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_11(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(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": 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 3, 2, 0], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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