01234_01243_01324_01342_01423_01432_02134_02143_02314_02341_02413_02431_03124_03142_03214_03241_03412_03421_04123_04132_04213_04231_04312_04321_12034_12043_12304_12340_12403_12430_13024_13042_13204_13240_13402_13420_14023_14032_14203_14230_14302_14320_23014_23041_23104_23140_23401_23410_24013_24031_24103_24130_24301_24310_34012_34021_34102_34120_34201_34210
Counting sequence:
1, 1, 2, 6, 24, 60, 120, 210, 336, 504, 720, 990, 1320, 1716, 2184, 2730, 3360, 4080, 4896, 5814, 6840, 7980, 9240, 10626, 12144, 13800, 15600, 17550, 19656, 21924, 24360, 26970, 29760, 32736, 35904, 39270, 42840, 46620, 50616, 54834, 59280, 63960, 68880, 74046, 79464, 85140, 91080, 97290, 103776, 110544, 117600, 124950, 132600, 140556, 148824, 157410, 166320, 175560, 185136, 195054, 205320, 215940, 226920, 238266, 249984, 262080, 274560, 287430, 300696, 314364, 328440, 342930, 357840, 373176, 388944, 405150, 421800, 438900, 456456, 474474, 492960, 511920, 531360, 551286, 571704, 592620, 614040, 635970, 658416, 681384, 704880, 728910, 753480, 778596, 804264, 830490, 857280, 884640, 912576, 941094, 970200
Generating function in Maple syntax:
(2*x^6-7*x^5+9*x^4+4*x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
\frac{2 x^{6}-7 x^{5}+9 x^{4}+4 x^{2}-3 x +1}{\left(x -1\right)^{4}}
Generating function in sympy syntax:
(2*x**6 - 7*x**5 + 9*x**4 + 4*x**2 - 3*x + 1)/(x - 1)**4
Implicit equation for the generating function in Maple syntax:
-(x-1)^4*F(x)+2*x^6-7*x^5+9*x^4+4*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{4} F \! \left(x \right)+2 x^{6}-7 x^{5}+9 x^{4}+4 x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n*(-1+n)*(-2+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ n \left(-1+n \right) \left(-2+n \right) & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 60
a(6) = 120
a(n) = n*(-1+n)*(-2+n), n >= 7
Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 60
a(6) = 120
a{\left(n \right)} = n \left(n - 2\right) \left(n - 1\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/24492/
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[23,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[16,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[1,x]+F[4,x]
F[15,x] = F[4,x]
F[16,x] = F[11,x]+F[17,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[4,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[15,x]
F[23,x] = F[2,x]+F[24,x]
F[24,x] = F[18,x]+F[25,x]+F[33,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[36,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[18,x]+F[29,x]+F[33,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[11,x]+F[17,x]
F[32,x] = F[17,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[16,x]+F[35,x]
F[35,x] = F[17,x]
F[36,x] = F[24,x]+F[37,x]
F[37,x] = F[18,x]+F[38,x]+F[44,x]+F[46,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[48,x]
F[40,x] = F[28,x]+F[41,x]
F[41,x] = F[18,x]+F[42,x]+F[44,x]+F[46,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[17,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[32,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[35,x]
F[48,x] = F[37,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[41,x]
F[53,x] = F[49,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_{23}\! \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_{16}\! \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_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \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_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)
F_{23}\! \left(x \right) = F_{2}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)+F_{33}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{36}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{32}\! \left(x \right) = F_{17}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{16}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{40}\! \left(x \right) = F_{28}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{17}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{32}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{35}\! \left(x \right)
F_{48}\! \left(x \right) = F_{37}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{41}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \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_23(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_16(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_4(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_1(x) + F_4(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_11(x) + F_17(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_4(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_15(x))
Eq(F_23(x), F_2(x) + F_24(x))
Eq(F_24(x), F_18(x) + F_25(x) + F_33(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_36(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_18(x) + F_29(x) + F_33(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_11(x) + F_17(x))
Eq(F_32(x), F_17(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_16(x) + F_35(x))
Eq(F_35(x), F_17(x))
Eq(F_36(x), F_24(x) + F_37(x))
Eq(F_37(x), F_18(x) + F_38(x) + F_44(x) + F_46(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_48(x))
Eq(F_40(x), F_28(x) + F_41(x))
Eq(F_41(x), F_18(x) + F_42(x) + F_44(x) + F_46(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_17(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_32(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_35(x))
Eq(F_48(x), F_37(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_41(x))
Eq(F_53(x), F_49(x))
Pack JSON:
{"name": "point_placements", "initial_strats": [{"class_module": "tilings.strategies.factor", "strategy_class": "FactorFactory", "interleaving": null, "unions": false, "ignore_parent": true, "workable": true, "tracked": false}, {"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "PointCorroborationFactory", "ignore_parent": true}], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "strategy_class": "RowColumnSeparationStrategy", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "strategy_class": "InsertionEncodingVerificationStrategy", "ignore_parent": false}, {"class_module": "tilings.strategies.verification", "strategy_class": "OneByOneVerificationStrategy", "ignore_parent": false, "basis": [], "symmetry": false}, {"class_module": "tilings.strategies.verification", "strategy_class": "LocallyFactorableVerificationStrategy", "ignore_parent": false, "basis": [], "symmetry": false}], "expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "CellInsertionFactory", "ignore_parent": false, "maxreqlen": 1, "extra_basis": [], "one_cell_only": false}, {"class_module": "tilings.strategies.requirement_placement", "strategy_class": "PatternPlacementFactory", "partial": false, "ignore_parent": false, "dirs": [0, 1, 2, 3], "point_only": false}]], "symmetries": [], "iterative": false}
Specification JSON:
{"root": {"class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 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