0123_0132_1302_2301
Counting sequence:
1, 1, 2, 6, 20, 62, 174, 450, 1096, 2558, 5794, 12850, 28076, 60686, 130134, 277378, 588432, 1243518, 2619466, 5502706, 11531636, 24114126, 50328062, 104853506, 218099160, 452979582, 939518194, 1946150450, 4026524476, 8321490958, 17179860134, 35433470210, 73014433056, 150323843326, 309237632154, 635655145458, 1305670042372, 2680059575758, 5497558120526, 11269994164866, 23089744161896, 47278999971326, 96757023219522, 197912092973106, 404620278993100, 826832744057102, 1688849860231414, 3448068464670850, 7036874417729456, 14355223812204158, 29273397577866474, 59672695062614770, 121597189938956436, 247697979505327566, 504403158265442974, 1026820715040417538, 2089670227099851512, 4251398048237686398, 8646911284551287186, 17582052945254347826, 35740566642812184156, 72634054790231283726, 147573952589676333382, 299759591197780130306, 608742554432415115840, 1235931852938539866622, 2508757194024498923834, 5091301364343836145650, 10330176681277348800036, 20955501267734050526158, 42501298345826806808814, 86183188312371025030530, 174727559866176872782216, 354177486215223390897406, 717799705396186072346594, 1454488876723850725677618, 2946756685310658613199852, 5969071234347231549959438, 12089258196146291746903446, 24480747847196240787635586, 49565958604199796162782416, 100340843028014221500435326, 203099537695257701350453642, 411034778668973919399909106, 831740963894864872197651380, 1682824740903563811190792142, 3404335108034795755972379454, 6886041468524927779126158850, 13926825441960528092614920216, 28163135893742401253954840958, 56945241807127492645359471154, 115128423653540365565618301490, 232732727385651491681035094396, 470417214928444504461666936846, 950737950171172051122527127014, 1921282940970910186643440509698, 3882179963198952542083653271392, 7843588088912169421760850778878, 15845632502852867518708789753306, 32008177655762792387791755612146, 64650180611639699476331863140676
Generating function in Maple syntax:
-(9*x^5-21*x^4+28*x^3-20*x^2+7*x-1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
-\frac{9 x^{5}-21 x^{4}+28 x^{3}-20 x^{2}+7 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-9*x**5 + 21*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^4*F(x)+9*x^5-21*x^4+28*x^3-20*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+9 x^{5}-21 x^{4}+28 x^{3}-20 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
2^(-1+n)*n+2^n-5/3*n-1/3*n^3
Explicit closed form in latex syntax:
2^{-1+n} n +2^{n}-\frac{5 n}{3}-\frac{n^{3}}{3}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(n+2) = -1/3*n^3+2*n^2-4*a(n)+4*a(n+1)-5/3*n+2, n >= 6
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) = 20
a \! \left(5\right) = 62
a \! \left(n +2\right) = -\frac{n^{3}}{3}+2 n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{5 n}{3}+2, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14860/
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[22,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[11,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = x
F[11,x] = F[12,x]+F[6,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[10,x]*F[15,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[10,x]
F[17,x] = F[12,x]
F[18,x] = F[10,x]*F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[13,x]+F[24,x]+F[48,x]
F[24,x] = F[10,x]*F[25,x]
F[25,x] = F[26,x]+F[32,x]
F[26,x] = F[27,x]+F[6,x]
F[27,x] = F[28,x]
F[28,x] = F[10,x]*F[29,x]
F[29,x] = F[16,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[28,x]
F[32,x] = F[23,x]+F[33,x]
F[33,x] = 2*F[13,x]+F[34,x]+F[38,x]
F[34,x] = F[10,x]*F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[27,x]
F[37,x] = F[33,x]
F[38,x] = F[10,x]*F[39,x]
F[39,x] = F[40,x]+F[44,x]
F[40,x] = F[41,x]
F[41,x] = F[13,x]+F[18,x]+F[42,x]
F[42,x] = F[10,x]*F[43,x]
F[43,x] = F[10,x]+F[41,x]
F[44,x] = F[45,x]
F[45,x] = 2*F[13,x]+F[38,x]+F[46,x]
F[46,x] = F[10,x]*F[47,x]
F[47,x] = F[31,x]+F[45,x]
F[48,x] = F[10,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[19,x]+F[41,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[13,x]+F[48,x]+F[53,x]
F[53,x] = F[10,x]*F[54,x]
F[54,x] = F[52,x]+F[6,x]
F[55,x] = F[13,x]+F[46,x]+F[56,x]+F[59,x]
F[56,x] = F[10,x]*F[57,x]
F[57,x] = F[40,x]+F[58,x]
F[58,x] = F[55,x]
F[59,x] = F[10,x]*F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[10,x]*F[62,x]
F[62,x] = F[19,x]+F[60,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_{22}\! \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_{11}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \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_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right)+F_{24}\! \left(x \right)+F_{48}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{32}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{6}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{28}\! \left(x \right)
F_{32}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{27}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right)+F_{18}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{38}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{31}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{19}\! \left(x \right)+F_{41}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{13}\! \left(x \right)+F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{52}\! \left(x \right)+F_{6}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right)+F_{46}\! \left(x \right)+F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{40}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{55}\! \left(x \right)
F_{59}\! \left(x \right) = F_{10}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{10}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{19}\! \left(x \right)+F_{60}\! \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_22(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_11(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_6(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_10(x)*F_15(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_10(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_10(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_1(x) + F_19(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_13(x) + F_24(x) + F_48(x))
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_32(x))
Eq(F_26(x), F_27(x) + F_6(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_10(x)*F_29(x))
Eq(F_29(x), F_16(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_28(x))
Eq(F_32(x), F_23(x) + F_33(x))
Eq(F_33(x), 2*F_13(x) + F_34(x) + F_38(x))
Eq(F_34(x), F_10(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_27(x))
Eq(F_37(x), F_33(x))
Eq(F_38(x), F_10(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_44(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_13(x) + F_18(x) + F_42(x))
Eq(F_42(x), F_10(x)*F_43(x))
Eq(F_43(x), F_10(x) + F_41(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), 2*F_13(x) + F_38(x) + F_46(x))
Eq(F_46(x), F_10(x)*F_47(x))
Eq(F_47(x), F_31(x) + F_45(x))
Eq(F_48(x), F_10(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_19(x) + F_41(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_13(x) + F_48(x) + F_53(x))
Eq(F_53(x), F_10(x)*F_54(x))
Eq(F_54(x), F_52(x) + F_6(x))
Eq(F_55(x), F_13(x) + F_46(x) + F_56(x) + F_59(x))
Eq(F_56(x), F_10(x)*F_57(x))
Eq(F_57(x), F_40(x) + F_58(x))
Eq(F_58(x), F_55(x))
Eq(F_59(x), F_10(x)*F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_10(x)*F_62(x))
Eq(F_62(x), F_19(x) + F_60(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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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