0123_0132_1203_2031_2310
Counting sequence:
1, 1, 2, 6, 19, 53, 124, 252, 460, 774, 1223, 1839, 2657, 3715, 5054, 6718, 8754, 11212, 14145, 17609, 21663, 26369, 31792, 38000, 45064, 53058, 62059, 72147, 83405, 95919, 109778, 125074, 141902, 160360, 180549, 202573, 226539, 252557, 280740, 311204, 344068, 379454, 417487, 458295, 502009, 548763, 598694, 651942, 708650, 768964, 833033, 901009, 973047, 1049305, 1129944, 1215128, 1305024, 1399802, 1499635, 1604699, 1715173, 1831239, 1953082, 2080890, 2214854, 2355168, 2502029, 2655637, 2816195, 2983909, 3158988, 3341644, 3532092, 3730550, 3937239, 4152383, 4376209, 4608947, 4850830, 5102094, 5362978, 5633724, 5914577, 6205785, 6507599, 6820273, 7144064, 7479232, 7826040, 8184754, 8555643, 8938979, 9335037, 9744095, 10166434, 10602338, 11052094, 11515992, 11994325, 12487389, 12995483
Generating function in Maple syntax:
(x^8+2*x^6-2*x^5-4*x^4+4*x^3-7*x^2+4*x-1)/(-1+x)^5
Generating function in latex syntax:
\frac{x^{8}+2 x^{6}-2 x^{5}-4 x^{4}+4 x^{3}-7 x^{2}+4 x -1}{\left(-1+x \right)^{5}}
Generating function in sympy syntax:
(x**8 + 2*x**6 - 2*x**5 - 4*x**4 + 4*x**3 - 7*x**2 + 4*x - 1)/(x - 1)**5
Implicit equation for the generating function in Maple syntax:
-(-1+x)^5*F(x)+x^8+2*x^6-2*x^5-4*x^4+4*x^3-7*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(-1+x \right)^{5} F \! \left(x \right)+x^{8}+2 x^{6}-2 x^{5}-4 x^{4}+4 x^{3}-7 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,-42-73/8*n^2+413/12*n+1/8*n^4+7/12*n^3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ -42-\frac{73}{8} n^{2}+\frac{413}{12} n +\frac{1}{8} n^{4}+\frac{7}{12} n^{3} & \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) = 124
a(7) = 252
a(8) = 460
a(n) = -42-73/8*n^2+413/12*n+1/8*n^4+7/12*n^3, n >= 9
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) = 124
a \! \left(7\right) = 252
a \! \left(8\right) = 460
a \! \left(n \right) = -42-\frac{73}{8} n^{2}+\frac{413}{12} n +\frac{1}{8} n^{4}+\frac{7}{12} n^{3}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16443/
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[21,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[10,x]*F[4,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[13,x]+F[23,x]+F[45,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[30,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[16,x]+F[29,x]
F[29,x] = F[27,x]
F[30,x] = F[22,x]+F[31,x]
F[31,x] = 2*F[13,x]+F[32,x]+F[36,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[26,x]
F[35,x] = F[31,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[44,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[1,x]+F[19,x]
F[44,x] = F[36,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[39,x]+F[41,x]
F[48,x] = F[49,x]+F[54,x]
F[49,x] = F[13,x]+F[45,x]+F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[7,x]
F[52,x] = F[53,x]
F[53,x] = F[16,x]*F[4,x]
F[54,x] = 2*F[13,x]+F[55,x]+F[58,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[38,x]+F[57,x]
F[57,x] = F[54,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[19,x]+F[62,x]
F[62,x] = F[53,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_{21}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right)+F_{23}\! \left(x \right)+F_{45}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{22}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{26}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{44}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \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_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \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_{39}\! \left(x \right)+F_{41}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right)+F_{45}\! \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_{7}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{38}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{54}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{19}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{53}\! \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_21(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_10(x)*F_4(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_13(x) + F_23(x) + F_45(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_30(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_16(x) + F_29(x))
Eq(F_29(x), F_27(x))
Eq(F_30(x), F_22(x) + F_31(x))
Eq(F_31(x), 2*F_13(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_26(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_44(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_1(x) + F_19(x))
Eq(F_44(x), F_36(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_39(x) + F_41(x))
Eq(F_48(x), F_49(x) + F_54(x))
Eq(F_49(x), F_13(x) + F_45(x) + F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_7(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_16(x)*F_4(x))
Eq(F_54(x), 2*F_13(x) + F_55(x) + F_58(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_38(x) + F_57(x))
Eq(F_57(x), F_54(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_19(x) + F_62(x))
Eq(F_62(x), F_53(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, 2, 0, 3], "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, 1, 0], "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, 2, 0, 3], "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, 1, 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