0132_1203_1302_2031_2301
Counting sequence:
1, 1, 2, 6, 19, 57, 159, 415, 1026, 2432, 5584, 12520, 27581, 59971, 129133, 276013, 586612, 1241138, 2616406, 5498830, 11526791, 24108141, 50320747, 104844651, 218088534, 452966932, 939503244, 1946132900, 4026504001, 8321467207, 17179832729, 35433438745, 73014397096, 150323802406, 309237585778, 635655093098, 1305669983467, 2680059509713, 5497558046711, 11269994082615, 23089744070506, 47278999870056, 96757023107592, 197912092849696, 404620278857349, 826832743908107, 1688849860068229, 3448068464492485, 7036874417534876, 14355223811992282, 29273397577636174, 59672695062364870, 121597189938685711, 247697979505034741, 504403158265126723, 1026820715040076483, 2089670227099484222, 4251398048237291388, 8646911284550862916, 17582052945253892700, 35740566642811696521, 72634054790230761871, 147573952589675775537, 299759591197779534641, 608742554432414480464, 1235931852938539189582, 2508757194024498203114, 5091301364343835379170, 10330176681277347985651, 20955501267734049661657, 42501298345826805891919, 86183188312371024058895, 174727559866176871753426, 354177486215223389808976, 717799705396186071195968, 1454488876723850724462168, 2946756685310658611916877, 5969071234347231548606163, 12089258196146291745477021, 24480747847196240786133085, 49565958604199796161200836, 100340843028014221498771586, 203099537695257701348704582, 411034778668973919398071486, 831740963894864872195721879, 1682824740903563811188767357, 3404335108034795755970255899, 6886041468524927779123932955, 13926825441960528092612588326, 28163135893742401253952399332, 56945241807127492645356915964, 115128423653540365565615628820, 232732727385651491681032300241, 470417214928444504461664017111, 950737950171172051122524077513, 1921282940970910186643437326153, 3882179963198952542083649949432, 7843588088912169421760847314038, 15845632502852867518708786141026, 32008177655762792387791751847770, 64650180611639699476331859219451
Generating function in Maple syntax:
-(5*x^6-26*x^5+48*x^4-48*x^3+27*x^2-8*x+1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
-\frac{5 x^{6}-26 x^{5}+48 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:
(-5*x**6 + 26*x**5 - 48*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)+5*x^6-26*x^5+48*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)+5 x^{6}-26 x^{5}+48 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
2^(-1+n)*n+2^n-1/24*n^4-1/12*n^3-11/24*n^2-17/12*n
Explicit closed form in latex syntax:
2^{-1+n} n +2^{n}-\frac{n^{4}}{24}-\frac{n^{3}}{12}-\frac{11 n^{2}}{24}-\frac{17 n}{12}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 57
a(6) = 159
a(n+2) = -4*a(n)+4*a(n+1)-1/24*(n+2)*(n^3-8*n^2+15*n-24), 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) = 57
a \! \left(6\right) = 159
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{\left(n +2\right) \left(n^{3}-8 n^{2}+15 n -24\right)}{24}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19894/
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[12,x]*F[4,x]
F[4,x] = F[28,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]+F[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[10,x]
F[19,x] = F[14,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[1,x]+F[22,x]
F[25,x] = F[15,x]+F[20,x]+F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[10,x]+F[25,x]
F[28,x] = F[2,x]+F[29,x]
F[29,x] = F[15,x]+F[30,x]+F[34,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[6,x]
F[33,x] = F[29,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[21,x]+F[36,x]
F[36,x] = F[37,x]+F[57,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[22,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[40,x]
F[43,x] = F[37,x]+F[44,x]
F[44,x] = 2*F[15,x]+F[45,x]+F[49,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[41,x]
F[48,x] = F[44,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[22,x]+F[51,x]
F[54,x] = 2*F[15,x]+F[49,x]+F[55,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[41,x]+F[54,x]
F[57,x] = 2*F[15,x]+F[58,x]+F[62,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[25,x]
F[61,x] = F[57,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[51,x]+F[64,x]
F[64,x] = 2*F[15,x]+F[62,x]+F[65,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[25,x]+F[64,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{28}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{57}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{22}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{41}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{22}\! \left(x \right)+F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{41}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{25}\! \left(x \right)
F_{61}\! \left(x \right) = F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{51}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{25}\! \left(x \right)+F_{64}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_28(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_10(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_1(x) + F_22(x))
Eq(F_25(x), F_15(x) + F_20(x) + F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_10(x) + F_25(x))
Eq(F_28(x), F_2(x) + F_29(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_34(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_6(x))
Eq(F_33(x), F_29(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_21(x) + F_36(x))
Eq(F_36(x), F_37(x) + F_57(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_22(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_40(x))
Eq(F_43(x), F_37(x) + F_44(x))
Eq(F_44(x), 2*F_15(x) + F_45(x) + F_49(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_41(x))
Eq(F_48(x), F_44(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_22(x) + F_51(x))
Eq(F_54(x), 2*F_15(x) + F_49(x) + F_55(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_41(x) + F_54(x))
Eq(F_57(x), 2*F_15(x) + F_58(x) + F_62(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_25(x))
Eq(F_61(x), F_57(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_51(x) + F_64(x))
Eq(F_64(x), 2*F_15(x) + F_62(x) + F_65(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_25(x) + F_64(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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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": [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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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": [2, 3, 0, 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