0123_0132_1032_2103_2130
Counting sequence:
1, 1, 2, 6, 19, 51, 129, 323, 797, 1953, 4761, 11569, 28051, 67917, 164283, 397127, 959579, 2317971, 5598249, 13518883, 32643157, 78816753, 190295361, 459437729, 1109219771, 2677956477, 6465260883, 15608685607, 37682967619, 90975163731, 219634173489, 530244932003, 1280126337197, 3090501327393, 7461135012681, 18012781094449, 43486712963971, 104986232526477, 253459219283403, 611904737863847, 1477268803048139, 3566442518767731, 8610154123428249, 20786751223276483, 50183657310478117, 121154067042381873, 292491793333887921, 706137656846952929, 1704767112103235051, 4115671889265659517, 9936110903922231843, 23987893718610037447, 57911898375929898739, 139811690526757341171, 337535279520519679329, 814882249715159304323, 1967299779189275990717, 4749481808479511592993, 11466263396772537186681, 27682008603034624283569, 66830280604476062081011, 161342569814631063089997, 389515420238016779232603, 940373410297587527171207, 2270262240844393330162619, 5480897892004498589700051, 13232058024882716408353929, 31945013941817381707402723, 77122085908594256022945397, 186189185759130120254074353, 449500457427055499231660961, 1085190100613566347918743969, 2619880658654714426971063451, 6324951417923846662964133117, 15269783494503785445904506483, 36864518406933646708881585127, 88998820308374685710781292579, 214862159023688854131666225171, 518723138355761836822449413649, 1252308435735227806626122778083, 3023340009826242171772588366157, 7298988455387752150718750632353, 17621316920601811195455434149161, 42541622296591479264422414570929, 102704561513784939169338403449571, 247950745324161631770930157268877, 598606052162108646324067793944683, 1445162849648379642199765756914407, 3488931751458869092117168395487019, 8423026352566119705608371647358131, 20334984456591111543901749877386489, 49092995265748347713153978688784003, 118520974988087814930519652728790597, 286134945241923990454245336906854193, 690790865471935816679372324776824081, 1667716676185795657533404037455316449, 4026224217843527186306956448916596171, 9720165111872850118428507035512462077, 23466554441589227566005936669394612803, 56653273995051305481563536623978733447, 136773102431691838903098132316482217939
Generating function in Maple syntax:
-(x+1)*(x^6+3*x^5-4*x^4-x^3-2*x^2+3*x-1)/(x^2+x-1)/(x^2+2*x-1)
Generating function in latex syntax:
-\frac{\left(x +1\right) \left(x^{6}+3 x^{5}-4 x^{4}-x^{3}-2 x^{2}+3 x -1\right)}{\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right)}
Generating function in sympy syntax:
(-x - 1)*(x**6 + 3*x**5 - 4*x**4 - x**3 - 2*x**2 + 3*x - 1)/((x**2 + x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(x^2+2*x-1)*F(x)+(x+1)*(x^6+3*x^5-4*x^4-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(x^{2}+2 x -1\right) F \! \left(x \right)+\left(x +1\right) \left(x^{6}+3 x^{5}-4 x^{4}-x^{3}-2 x^{2}+3 x -1\right) = 0
Explicit closed form in Maple syntax:
-piecewise(n = 0,10,n = 1,-4,Or(n = 2,n = 3),1,0)+(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-11/2*(2^(1/2)-1)^(-n)*2^(1/2)+11/2*(-1-2^(1/2))^(-n)*2^(1/2)-(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-3*(1/2*5^(1/2)-1/2)^(-n)+17/2*(2^(1/2)-1)^(-n)+17/2*(-1-2^(1/2))^(-n)-3*(-1/2*5^(1/2)-1/2)^(-n)
Explicit closed form in latex syntax:
-\left(\left\{\begin{array}{cc}10 & n =0 \\ -4 & n =1 \\ 1 & n =2\text{ or } n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)+\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}-\frac{11 \left(\sqrt{2}-1\right)^{-n} \sqrt{2}}{2}+\frac{11 \left(-1-\sqrt{2}\right)^{-n} \sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}-3 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+\frac{17 \left(\sqrt{2}-1\right)^{-n}}{2}+\frac{17 \left(-1-\sqrt{2}\right)^{-n}}{2}-3 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 129
a(7) = 323
a(n+1) = -1/3*a(n)+a(n+3)-1/3*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) = 51
a \! \left(6\right) = 129
a \! \left(7\right) = 323
a \! \left(n +1\right) = -\frac{a \! \left(n \right)}{3}+a \! \left(n +3\right)-\frac{a \! \left(n +4\right)}{3}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16381/
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[18,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[17,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]
F[16,x] = F[4,x]
F[17,x] = F[4,x]*F[7,x]
F[18,x] = F[19,x]+F[2,x]
F[19,x] = F[13,x]+F[20,x]+F[27,x]
F[20,x] = F[21,x]*F[4,x]
F[21,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[4,x]
F[25,x] = F[13,x]+F[14,x]+F[26,x]
F[26,x] = F[11,x]*F[4,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[35,x]
F[29,x] = F[2,x]+F[30,x]
F[30,x] = F[13,x]+F[31,x]+F[34,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]+F[4,x]
F[34,x] = F[2,x]*F[4,x]
F[35,x] = F[19,x]+F[36,x]
F[36,x] = F[13,x]+F[37,x]+F[40,x]+F[41,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[16,x]*F[4,x]
F[40,x] = 0
F[41,x] = F[19,x]*F[4,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_{18}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{2}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right)+F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \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_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{35}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right)+F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{19}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right)+F_{37}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = 0
F_{41}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \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_18(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_17(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_4(x)*F_7(x))
Eq(F_18(x), F_19(x) + F_2(x))
Eq(F_19(x), F_13(x) + F_20(x) + F_27(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(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_4(x))
Eq(F_25(x), F_13(x) + F_14(x) + F_26(x))
Eq(F_26(x), F_11(x)*F_4(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_35(x))
Eq(F_29(x), F_2(x) + F_30(x))
Eq(F_30(x), F_13(x) + F_31(x) + F_34(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x) + F_4(x))
Eq(F_34(x), F_2(x)*F_4(x))
Eq(F_35(x), F_19(x) + F_36(x))
Eq(F_36(x), F_13(x) + F_37(x) + F_40(x) + F_41(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_16(x)*F_4(x))
Eq(F_40(x), 0)
Eq(F_41(x), F_19(x)*F_4(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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 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": [[{"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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 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": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": 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