0132_0231_1032_1203_2301
Counting sequence:
1, 1, 2, 6, 19, 53, 132, 305, 674, 1454, 3101, 6584, 13961, 29595, 62714, 132787, 280788, 592748, 1248907, 2626114, 5510819, 11541441, 24125872, 50342021, 104869974, 218118458, 453002057, 939544220, 1946180429, 4026558839, 8321530166, 17179904679, 35433520616, 73014489880, 150323907159, 309237703622, 635655225223, 1305670131133, 2680059674252, 5497558229529, 11269994285194, 23089744294406, 47279000116917, 96757023379136, 197912093147729, 404620279183763, 826832744264882, 1688849860457435, 3448068464916284, 7036874417995524, 14355223812492131, 29273397578177674, 59672695062950571, 121597189939318265, 247697979505716904, 504403158265861357, 1026820715040866558, 2089670227100332818, 4251398048238201697, 8646911284551838244, 17582052945254936469, 35740566642812812271, 72634054790231953262, 147573952589677046351, 299759591197780888784, 608742554432415921968, 1235931852938540722607, 2508757194024499831950, 5091301364343837108239, 10330176681277349819509, 20955501267734051604996, 42501298345826807949569, 86183188312371026235826, 174727559866176874054750, 354177486215223392239949, 717799705396186073761992, 1454488876723850727168793, 2946756685310658614769803, 5969071234347231551611242, 12089258196146291748640259, 24480747847196240789460644, 49565958604199796164699036, 100340843028014221502446907, 203099537695257701352563666, 411034778668973919402121139, 831740963894864872199969073, 1682824740903563811193219232, 3404335108034795755974919765, 6886041468524927779128816294, 13926825441960528092617698794, 28163135893742401253957744761, 56945241807127492645362504364, 115128423653540365565621468381, 232732727385651491681038399335, 470417214928444504461670384294, 950737950171172051122530721527, 1921282940970910186643444255928, 3882179963198952542083657174088, 7843588088912169421760854842887, 15845632502852867518708793983574, 32008177655762792387791760013719
Generating function in Maple syntax:
(3*x^7-14*x^6+30*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{3 x^{7}-14 x^{6}+30 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:
(3*x**7 - 14*x**6 + 30*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)-3*x^7+14*x^6-30*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)-3 x^{7}+14 x^{6}-30 x^{5}+48 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/24*(6*n+6)*2^n+1/24*n^4-1/12*n^3-13/24*n^2+7/12*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(6 n +6\right) 2^{n}}{24}+\frac{n^{4}}{24}-\frac{n^{3}}{12}-\frac{13 n^{2}}{24}+\frac{7 n}{12} & \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) = 53
a(6) = 132
a(7) = 305
a(n+2) = 1/24*n^4-5/12*n^3-1/24*n^2+4*a(n+1)-4*a(n)+41/12*n-1, 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) = 53
a \! \left(6\right) = 132
a \! \left(7\right) = 305
a \! \left(n +2\right) = \frac{n^{4}}{24}-\frac{5 n^{3}}{12}-\frac{n^{2}}{24}+4 a \! \left(n +1\right)-4 a \! \left(n \right)+\frac{41 n}{12}-1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18854/
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[22,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[18,x]+F[24,x]+F[40,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[31,x]
F[26,x] = F[10,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[14,x]+F[30,x]
F[30,x] = F[28,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[18,x]+F[21,x]+F[24,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[14,x]+F[36,x]
F[39,x] = F[34,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[13,x]+F[42,x]
F[42,x] = F[43,x]+F[50,x]
F[43,x] = F[18,x]+F[44,x]+F[48,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[12,x]+F[46,x]
F[46,x] = F[18,x]+F[44,x]+F[47,x]
F[47,x] = F[12,x]*F[14,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[14,x]+F[43,x]
F[50,x] = F[18,x]+F[51,x]+F[52,x]+F[54,x]
F[51,x] = 0
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[17,x]+F[50,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[36,x]+F[56,x]
F[56,x] = 2*F[18,x]+F[54,x]+F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[17,x]+F[56,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_{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_{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_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)+F_{40}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \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_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{34}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{50}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{14}\! \left(x \right)+F_{43}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = 0
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{17}\! \left(x \right)+F_{50}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{36}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{17}\! \left(x \right)+F_{56}\! \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_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_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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_18(x) + F_24(x) + F_40(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_31(x))
Eq(F_26(x), F_10(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_14(x) + F_30(x))
Eq(F_30(x), F_28(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_18(x) + F_21(x) + F_24(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_14(x) + F_36(x))
Eq(F_39(x), F_34(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_13(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_50(x))
Eq(F_43(x), F_18(x) + F_44(x) + F_48(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_12(x) + F_46(x))
Eq(F_46(x), F_18(x) + F_44(x) + F_47(x))
Eq(F_47(x), F_12(x)*F_14(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_14(x) + F_43(x))
Eq(F_50(x), F_18(x) + F_51(x) + F_52(x) + F_54(x))
Eq(F_51(x), 0)
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_17(x) + F_50(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_36(x) + F_56(x))
Eq(F_56(x), 2*F_18(x) + F_54(x) + F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_17(x) + F_56(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": [0, 2, 3, 1], "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, 2, 0, 3], "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": [0, 2, 3, 1], "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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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