0123_0132_1320_2301
Counting sequence:
1, 1, 2, 6, 20, 60, 153, 340, 679, 1248, 2148, 3506, 5478, 8252, 12051, 17136, 23809, 32416, 43350, 57054, 74024, 94812, 120029, 150348, 186507, 229312, 279640, 338442, 406746, 485660, 576375, 680168, 798405, 932544, 1084138, 1254838, 1446396, 1660668, 1899617, 2165316, 2459951, 2785824, 3145356, 3541090, 3975694, 4451964, 4972827, 5541344, 6160713, 6834272, 7565502, 8358030, 9215632, 10142236, 11141925, 12218940, 13377683, 14622720, 15958784, 17390778, 18923778, 20563036, 22313983, 24182232, 26173581, 28294016, 30549714, 32947046, 35492580, 38193084, 41055529, 44087092, 47295159, 50687328, 54271412, 58055442, 62047670, 66256572, 70690851, 75359440, 80271505, 85436448, 90863910, 96563774, 102546168, 108821468, 115400301, 122293548, 129512347, 137068096, 144972456, 153237354, 161874986, 170897820, 180318599, 190150344, 200406357, 211100224, 222245818, 233857302, 245949132
Generating function in Maple syntax:
(x^7-2*x^6-x^5+9*x^4-11*x^3+11*x^2-5*x+1)/(x-1)^6
Generating function in latex syntax:
\frac{x^{7}-2 x^{6}-x^{5}+9 x^{4}-11 x^{3}+11 x^{2}-5 x +1}{\left(x -1\right)^{6}}
Generating function in sympy syntax:
(x**7 - 2*x**6 - x**5 + 9*x**4 - 11*x**3 + 11*x**2 - 5*x + 1)/(x - 1)**6
Implicit equation for the generating function in Maple syntax:
-(x-1)^6*F(x)+x^7-2*x^6-x^5+9*x^4-11*x^3+11*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{6} F \! \left(x \right)+x^{7}-2 x^{6}-x^{5}+9 x^{4}-11 x^{3}+11 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-3+1/40*n^5-1/24*n^4+1/8*n^3-23/24*n^2+77/20*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -3+\frac{1}{40} n^{5}-\frac{1}{24} n^{4}+\frac{1}{8} n^{3}-\frac{23}{24} n^{2}+\frac{77}{20} n & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 60
a(6) = 153
a(7) = 340
a(n) = 1/120*(-1+n)*(3*n^4-2*n^3+13*n^2-102*n+360), 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) = 20
a \! \left(5\right) = 60
a \! \left(6\right) = 153
a \! \left(7\right) = 340
a \! \left(n \right) = \frac{\left(-1+n \right) \left(3 n^{4}-2 n^{3}+13 n^{2}-102 n +360\right)}{120}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14869/
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[22,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[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,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[60,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[38,x]
F[26,x] = F[27,x]+F[7,x]
F[27,x] = F[13,x]+F[14,x]+F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[34,x]
F[30,x] = F[19,x]+F[31,x]
F[31,x] = F[13,x]+F[18,x]+F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[31,x]+F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[4,x]
F[37,x] = F[35,x]
F[38,x] = F[23,x]+F[39,x]
F[39,x] = F[13,x]+F[40,x]+F[53,x]+F[59,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[47,x]
F[42,x] = F[27,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[31,x]
F[47,x] = F[39,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[43,x]
F[52,x] = F[48,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[46,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[12,x]+F[56,x]
F[59,x] = 0
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[30,x]+F[62,x]
F[62,x] = F[56,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[63,x]+F[7,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_{22}\! \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_{18}\! \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_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \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_{60}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{38}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{30}\! \left(x \right) = F_{19}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{18}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{31}\! \left(x \right)+F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{23}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \left(x \right)+F_{40}\! \left(x \right)+F_{53}\! \left(x \right)+F_{59}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{27}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{31}\! \left(x \right)
F_{47}\! \left(x \right) = F_{39}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{43}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{46}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = 0
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{30}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{56}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{63}\! \left(x \right)+F_{7}\! \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_22(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_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(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_60(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_38(x))
Eq(F_26(x), F_27(x) + F_7(x))
Eq(F_27(x), F_13(x) + F_14(x) + F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_34(x))
Eq(F_30(x), F_19(x) + F_31(x))
Eq(F_31(x), F_13(x) + F_18(x) + F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_31(x) + F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_4(x))
Eq(F_37(x), F_35(x))
Eq(F_38(x), F_23(x) + F_39(x))
Eq(F_39(x), F_13(x) + F_40(x) + F_53(x) + F_59(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_47(x))
Eq(F_42(x), F_27(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_31(x))
Eq(F_47(x), F_39(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_43(x))
Eq(F_52(x), F_48(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_46(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_12(x) + F_56(x))
Eq(F_59(x), 0)
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_30(x) + F_62(x))
Eq(F_62(x), F_56(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_63(x) + F_7(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, 2, 0], "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, 2, 0], "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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