0213_0231_0312_1032_1302_2031_3120
Counting sequence:
1, 1, 2, 6, 17, 44, 108, 255, 584, 1307, 2877, 6260, 13511, 28990, 61922, 131773, 279514, 591173, 1246987, 2623802, 5508065, 11538192, 24122072, 50337611, 104864892, 218112639, 452995433, 939536720, 1946171979, 4026549362, 8321519582, 17179892905, 35433507566, 73014475465, 150323891287, 309237686198, 635655206149, 1305670110308, 2680059651572, 5497558204887, 11269994258480, 23089744265507, 47279000085717, 96757023345516, 197912093111567, 404620279144934, 826832744223258, 1688849860412885, 3448068464868674, 7036874417944717, 14355223812437987, 29273397578120050, 59672695062889321, 121597189939253240, 247697979505647952, 504403158265788323, 1026820715040789284, 2089670227100251143, 4251398048238115457, 8646911284551747272, 17582052945254840595, 35740566642812711322, 72634054790231847062, 147573952589676934721, 299759591197780771542, 608742554432415798929, 1235931852938540593583, 2508757194024499696750, 5091301364343836966669, 10330176681277349671372, 20955501267734051450092, 42501298345826807787695, 86183188312371026066776, 174727559866176873878315, 354177486215223392055917, 717799705396186073570148, 1454488876723850726968919, 2946756685310658614561678, 5969071234347231551394642, 12089258196146291748414957, 24480747847196240789226410, 49565958604199796164455637, 100340843028014221502194107, 203099537695257701352301226, 411034778668973919401848817, 831740963894864872199686624, 1682824740903563811192926408, 3404335108034795755974616315, 6886041468524927779128501964, 13926825441960528092617373327, 28163135893742401253957407897, 56945241807127492645362155840, 115128423653540365565621107931, 232732727385651491681038026690, 470417214928444504461669999182, 950737950171172051122530323673, 1921282940970910186643443845054, 3882179963198952542083656749913, 7843588088912169421760854405127, 15845632502852867518708793531942, 32008177655762792387791759547925
Generating function in Maple syntax:
-(4*x^8-3*x^7+3*x^6-21*x^5+46*x^4-48*x^3+27*x^2-8*x+1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
-\frac{4 x^{8}-3 x^{7}+3 x^{6}-21 x^{5}+46 x^{4}-48 x^{3}+27 x^{2}-8 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-4*x**8 + 3*x**7 - 3*x**6 + 21*x**5 - 46*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^5*F(x)+4*x^8-3*x^7+3*x^6-21*x^5+46*x^4-48*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+4 x^{8}-3 x^{7}+3 x^{6}-21 x^{5}+46 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/24*(6*n+6)*2^n+1/24*n^4-7/12*n^3+71/24*n^2-89/12*n+6)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(6 n +6\right) 2^{n}}{24}+\frac{n^{4}}{24}-\frac{7 n^{3}}{12}+\frac{71 n^{2}}{24}-\frac{89 n}{12}+6 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(5) = 44
a(6) = 108
a(7) = 255
a(8) = 584
a(n+2) = -4*a(n)+4*a(n+1)+1/24*(n-3)*(n-4)*(n^2-15*n+38), n >= 9
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) = 17
a \! \left(5\right) = 44
a \! \left(6\right) = 108
a \! \left(7\right) = 255
a \! \left(8\right) = 584
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{\left(n -3\right) \left(n -4\right) \left(n^{2}-15 n +38\right)}{24}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/23079/
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[8,x]
F[4,x] = F[35,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[18,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[8,x]
F[17,x] = F[1,x]+F[15,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[29,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[8,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[8,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[20,x]+F[21,x]+F[27,x]
F[27,x] = F[28,x]*F[8,x]
F[28,x] = F[15,x]+F[26,x]
F[29,x] = F[30,x]*F[8,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[8,x]
F[34,x] = F[15,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[8,x]
F[37,x] = F[38,x]+F[52,x]
F[38,x] = F[39,x]+F[51,x]
F[39,x] = F[40,x]+F[49,x]
F[40,x] = F[41,x]*F[45,x]
F[41,x] = F[1,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[17,x]*F[41,x]*F[8,x]
F[45,x] = F[1,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[17,x]*F[45,x]*F[8,x]
F[49,x] = F[50,x]
F[50,x] = F[25,x]*F[43,x]*F[47,x]*F[8,x]
F[51,x] = F[17,x]*F[45,x]*F[6,x]
F[52,x] = F[15,x]*F[25,x]*F[5,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{35}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \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_{21}\! \left(x \right)+F_{29}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{8}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{8}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{8}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{52}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{51}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right) F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{1}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right) F_{41}\! \left(x \right) F_{8}\! \left(x \right)
F_{45}\! \left(x \right) = F_{1}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{17}\! \left(x \right) F_{45}\! \left(x \right) F_{8}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{25}\! \left(x \right) F_{43}\! \left(x \right) F_{47}\! \left(x \right) F_{8}\! \left(x \right)
F_{51}\! \left(x \right) = F_{17}\! \left(x \right) F_{45}\! \left(x \right) F_{6}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right) F_{25}\! \left(x \right) F_{5}\! \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_8(x))
Eq(F_4(x), F_35(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_18(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_8(x))
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_8(x))
Eq(F_17(x), F_1(x) + F_15(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_29(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_8(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_8(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_20(x) + F_21(x) + F_27(x))
Eq(F_27(x), F_28(x)*F_8(x))
Eq(F_28(x), F_15(x) + F_26(x))
Eq(F_29(x), F_30(x)*F_8(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_8(x))
Eq(F_34(x), F_15(x) + F_32(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_8(x))
Eq(F_37(x), F_38(x) + F_52(x))
Eq(F_38(x), F_39(x) + F_51(x))
Eq(F_39(x), F_40(x) + F_49(x))
Eq(F_40(x), F_41(x)*F_45(x))
Eq(F_41(x), F_1(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_17(x)*F_41(x)*F_8(x))
Eq(F_45(x), F_1(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_17(x)*F_45(x)*F_8(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_25(x)*F_43(x)*F_47(x)*F_8(x))
Eq(F_51(x), F_17(x)*F_45(x)*F_6(x))
Eq(F_52(x), F_15(x)*F_25(x)*F_5(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": [0, 3, 1, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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, 1, 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