0123_0132_2301_2310_3201
Counting sequence:
1, 1, 2, 6, 19, 47, 77, 107, 137, 167, 197, 227, 257, 287, 317, 347, 377, 407, 437, 467, 497, 527, 557, 587, 617, 647, 677, 707, 737, 767, 797, 827, 857, 887, 917, 947, 977, 1007, 1037, 1067, 1097, 1127, 1157, 1187, 1217, 1247, 1277, 1307, 1337, 1367, 1397, 1427, 1457, 1487, 1517, 1547, 1577, 1607, 1637, 1667, 1697, 1727, 1757, 1787, 1817, 1847, 1877, 1907, 1937, 1967, 1997, 2027, 2057, 2087, 2117, 2147, 2177, 2207, 2237, 2267, 2297, 2327, 2357, 2387, 2417, 2447, 2477, 2507, 2537, 2567, 2597, 2627, 2657, 2687, 2717, 2747, 2777, 2807, 2837, 2867, 2897
Generating function in Maple syntax:
(2*x^6+15*x^5+9*x^4+3*x^3+x^2-x+1)/(x-1)^2
Generating function in latex syntax:
\frac{2 x^{6}+15 x^{5}+9 x^{4}+3 x^{3}+x^{2}-x +1}{\left(x -1\right)^{2}}
Generating function in sympy syntax:
(2*x**6 + 15*x**5 + 9*x**4 + 3*x**3 + x**2 - x + 1)/(x - 1)**2
Implicit equation for the generating function in Maple syntax:
-(x-1)^2*F(x)+2*x^6+15*x^5+9*x^4+3*x^3+x^2-x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{2} F \! \left(x \right)+2 x^{6}+15 x^{5}+9 x^{4}+3 x^{3}+x^{2}-x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,n = 4,19,-103+30*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 19 & n =4 \\ -103+30 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) = 19
a(5) = 47
a(6) = 77
a(n) = -103+30*n, 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) = 47
a \! \left(6\right) = 77
a \! \left(n \right) = -103+30 n, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16674/
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[29,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[24,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[16,x]+F[19,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[1,x]+F[16,x]
F[19,x] = F[20,x]+F[21,x]+F[23,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[19,x]+F[4,x]
F[23,x] = x^2
F[24,x] = F[20,x]+F[23,x]+F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[4,x]
F[28,x] = F[19,x]
F[29,x] = F[30,x]+F[50,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[1,x]+F[12,x]
F[34,x] = F[16,x]+F[35,x]
F[35,x] = F[20,x]+F[36,x]+F[38,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[12,x]+F[35,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[4,x]+F[41,x]
F[41,x] = F[20,x]+F[23,x]+F[42,x]
F[42,x] = x^2
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[4,x]+F[44,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[41,x]+F[47,x]
F[50,x] = F[20,x]+F[51,x]+F[68,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[64,x]
F[53,x] = F[54,x]+F[7,x]
F[54,x] = F[20,x]+F[25,x]+F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[40,x]+F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[20,x]+F[38,x]+F[42,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[41,x]
F[63,x] = F[47,x]
F[64,x] = F[65,x]+F[76,x]
F[65,x] = F[20,x]+F[66,x]+F[68,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[65,x]+F[7,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[40,x]+F[70,x]
F[70,x] = F[71,x]+F[73,x]
F[71,x] = F[20,x]+F[38,x]+F[72,x]
F[72,x] = F[12,x]*F[4,x]
F[73,x] = F[20,x]+F[60,x]+F[74,x]+F[75,x]
F[74,x] = F[24,x]*F[4,x]
F[75,x] = 0
F[76,x] = F[20,x]+F[60,x]+F[77,x]+F[79,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[54,x]+F[76,x]
F[79,x] = 0
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_{29}\! \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_{24}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = x^{2}
F_{24}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \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_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{19}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{50}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{34}\! \left(x \right) = F_{16}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{20}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right)+F_{35}\! \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_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = x^{2}
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{41}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{20}\! \left(x \right)+F_{51}\! \left(x \right)+F_{68}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{64}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{7}\! \left(x \right)
F_{54}\! \left(x \right) = F_{20}\! \left(x \right)+F_{25}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{40}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \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_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{41}\! \left(x \right)
F_{63}\! \left(x \right) = F_{47}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{76}\! \left(x \right)
F_{65}\! \left(x \right) = F_{20}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{65}\! \left(x \right)+F_{7}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{40}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{73}\! \left(x \right) = F_{20}\! \left(x \right)+F_{60}\! \left(x \right)+F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{75}\! \left(x \right) = 0
F_{76}\! \left(x \right) = F_{20}\! \left(x \right)+F_{60}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{54}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = 0
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_29(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_24(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_16(x) + F_19(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_1(x) + F_16(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_23(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_19(x) + F_4(x))
Eq(F_23(x), x**2)
Eq(F_24(x), F_20(x) + F_23(x) + F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_4(x))
Eq(F_28(x), F_19(x))
Eq(F_29(x), F_30(x) + F_50(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_1(x) + F_12(x))
Eq(F_34(x), F_16(x) + F_35(x))
Eq(F_35(x), F_20(x) + F_36(x) + F_38(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_12(x) + F_35(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_4(x) + F_41(x))
Eq(F_41(x), F_20(x) + F_23(x) + F_42(x))
Eq(F_42(x), x**2)
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_4(x) + F_44(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_41(x) + F_47(x))
Eq(F_50(x), F_20(x) + F_51(x) + F_68(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_64(x))
Eq(F_53(x), F_54(x) + F_7(x))
Eq(F_54(x), F_20(x) + F_25(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_40(x) + F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_20(x) + F_38(x) + F_42(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_41(x))
Eq(F_63(x), F_47(x))
Eq(F_64(x), F_65(x) + F_76(x))
Eq(F_65(x), F_20(x) + F_66(x) + F_68(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_65(x) + F_7(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_40(x) + F_70(x))
Eq(F_70(x), F_71(x) + F_73(x))
Eq(F_71(x), F_20(x) + F_38(x) + F_72(x))
Eq(F_72(x), F_12(x)*F_4(x))
Eq(F_73(x), F_20(x) + F_60(x) + F_74(x) + F_75(x))
Eq(F_74(x), F_24(x)*F_4(x))
Eq(F_75(x), 0)
Eq(F_76(x), F_20(x) + F_60(x) + F_77(x) + F_79(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_54(x) + F_76(x))
Eq(F_79(x), 0)
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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 2, 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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 2, 0, 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