0213_1230_1302_1320_3021
Counting sequence:
1, 1, 2, 6, 19, 55, 152, 410, 1092, 2888, 7607, 19989, 52450, 137506, 360302, 943778, 2471643, 6472139, 16946372, 44369562, 116166496, 304136692, 796254527, 2084644601, 5457707934, 14288525570, 37407943802, 97935427230, 256398534307, 671260493503, 1757383460432, 4600890719834, 12045290045340, 31534981594496, 82559658262727, 216143998896573, 565872347654458, 1481473058997154, 3878546853494822, 10154167540575482, 26583955831477611, 69597700056191507, 182209144502677052, 477029733719753946, 1248880057090079224, 3269610438251892460, 8559951258800501327, 22410243339985923425, 58670778764128484022, 153602092957207055618, 402135500115271424882, 1052804407401193488054, 2756277722108674050355, 7216028758957779943111, 18891808554817982070152, 49459396905582433838618, 129486382162068903308148, 338999749580850127519544, 887512866580846914546647, 2323538850162281902850277, 6083103683906955516030226, 15925772201560132653996322, 41694212920775947176740702, 109156866560771761615763666, 285776386761545895140870139, 748172293723876534016704475, 1958740494410100874589420852, 5128049189506453867641593370, 13425407074109305673905572112, 35148172032821535877535371108, 92019109024355419627731002207, 240909155040244913398148344649, 630708356096379628628235201870, 1651215913248894470940569140226, 4322939383650304590709005268202, 11317602237702020606155991593038, 29629867329455759339244047488963, 77571999750665260828030773780559, 203086131922540028672787974737472, 531686396016954834134727474223322, 1391973056128324488203728472608716, 3644232772368018653893186292070512, 9540725260975731511364892776746727, 24977943010559175941507282759781261, 65393103770701796412351808597352554, 171201368301546213456048786848643490, 448211001133936844215490048859700502, 1173431635100264319610617500457947690, 3072083904166856115296254090152754827, 8042820077400304027378232548366418723, 21056376328034055968618422970951215532, 55126308906701863881357103558858043994, 144322550392071535680112934315998446760, 377841342269512743166521813193883642716, 989201476416466693831652665680774358127, 2589763086979887338348176458068307654833, 6780087784523195321244817143159138706278, 17750500266589698625437955680263966787074, 46471413015245900555152671041122610077922, 121663738779148003040155359295448570192742, 318519803322198108565532329841057655669331
Generating function in Maple syntax:
-(x^6+2*x^5+x^4-2*x^3-4*x^2+4*x-1)/(x-1)/(x^2-3*x+1)/(x^2+x-1)
Generating function in latex syntax:
-\frac{x^{6}+2 x^{5}+x^{4}-2 x^{3}-4 x^{2}+4 x -1}{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right)}
Generating function in sympy syntax:
(-x**6 - 2*x**5 - x**4 + 2*x**3 + 4*x**2 - 4*x + 1)/((x - 1)*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2-3*x+1)*(x^2+x-1)*F(x)+x^6+2*x^5+x^4-2*x^3-4*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+x^{6}+2 x^{5}+x^{4}-2 x^{3}-4 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,-7/5*(3/2-1/2*5^(1/2))^(1-n)-7/5*(3/2+1/2*5^(1/2))^(1-n)-7/5*(-1/2*5^(1/2)-1/2)^(1-n)-7/5*(1/2*5^(1/2)-1/2)^(1-n)+1/20*(-43*5^(1/2)+117)*(3/2-1/2*5^(1/2))^(-n)+1/20*(43*5^(1/2)+117)*(3/2+1/2*5^(1/2))^(-n)+1/20*(-11*5^(1/2)-19)*(-1/2*5^(1/2)-1/2)^(-n)-1+1/20*(11*5^(1/2)-19)*(1/2*5^(1/2)-1/2)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ -\frac{7 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}}{5}-\frac{7 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n}}{5}-\frac{7 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{1-n}}{5}-\frac{7 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{1-n}}{5}\\+\frac{\left(-43 \sqrt{5}+117\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(43 \sqrt{5}+117\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\\\frac{\left(-11 \sqrt{5}-19\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}-1+\frac{\left(11 \sqrt{5}-19\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 152
a(n+4) = a(n)-2*a(n+1)-3*a(n+2)+4*a(n+3)-1, 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) = 152
a \! \left(n +4\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right)-1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20572/
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[14,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[18,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[14,x]
F[12,x] = F[13,x]+F[15,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = x
F[15,x] = F[10,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[10,x]*F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[14,x]*F[20,x]
F[20,x] = F[9,x]
F[21,x] = F[22,x]+F[33,x]
F[22,x] = F[23,x]
F[23,x] = F[14,x]*F[24,x]
F[24,x] = F[13,x]+F[25,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[32,x]
F[27,x] = 0
F[28,x] = F[14,x]*F[29,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[14,x]
F[31,x] = F[26,x]
F[32,x] = F[14,x]*F[22,x]
F[33,x] = F[27,x]+F[34,x]+F[46,x]
F[34,x] = F[14,x]*F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]+F[6,x]
F[37,x] = F[38,x]
F[38,x] = F[14,x]*F[6,x]
F[39,x] = F[33,x]+F[40,x]
F[40,x] = 2*F[27,x]+F[41,x]+F[45,x]
F[41,x] = F[14,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[37,x]
F[44,x] = F[40,x]
F[45,x] = F[14,x]*F[33,x]
F[46,x] = F[14,x]*F[47,x]
F[47,x] = F[21,x]+F[48,x]
F[48,x] = F[49,x]+F[53,x]
F[49,x] = F[50,x]
F[50,x] = F[14,x]*F[51,x]
F[51,x] = F[30,x]+F[52,x]
F[52,x] = F[50,x]
F[53,x] = F[54,x]
F[54,x] = F[14,x]*F[55,x]
F[55,x] = F[48,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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \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_{14}\! \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_{18}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{15}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{10}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{9}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{33}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{14}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)
F_{31}\! \left(x \right) = F_{26}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{22}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)+F_{46}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{6}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right) F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \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_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{21}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{53}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{14}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{30}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{14}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{48}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_21(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_14(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_18(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_14(x))
Eq(F_12(x), F_13(x) + F_15(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_10(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_10(x)*F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_14(x)*F_20(x))
Eq(F_20(x), F_9(x))
Eq(F_21(x), F_22(x) + F_33(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_14(x)*F_24(x))
Eq(F_24(x), F_13(x) + F_25(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_32(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_14(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_14(x))
Eq(F_31(x), F_26(x))
Eq(F_32(x), F_14(x)*F_22(x))
Eq(F_33(x), F_27(x) + F_34(x) + F_46(x))
Eq(F_34(x), F_14(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x) + F_6(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_14(x)*F_6(x))
Eq(F_39(x), F_33(x) + F_40(x))
Eq(F_40(x), 2*F_27(x) + F_41(x) + F_45(x))
Eq(F_41(x), F_14(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_37(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), F_14(x)*F_33(x))
Eq(F_46(x), F_14(x)*F_47(x))
Eq(F_47(x), F_21(x) + F_48(x))
Eq(F_48(x), F_49(x) + F_53(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_14(x)*F_51(x))
Eq(F_51(x), F_30(x) + F_52(x))
Eq(F_52(x), F_50(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_14(x)*F_55(x))
Eq(F_55(x), F_48(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": [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]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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": [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]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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