0123_0132_1032_1230_2310
Counting sequence:
1, 1, 2, 6, 19, 51, 106, 180, 273, 385, 516, 666, 835, 1023, 1230, 1456, 1701, 1965, 2248, 2550, 2871, 3211, 3570, 3948, 4345, 4761, 5196, 5650, 6123, 6615, 7126, 7656, 8205, 8773, 9360, 9966, 10591, 11235, 11898, 12580, 13281, 14001, 14740, 15498, 16275, 17071, 17886, 18720, 19573, 20445, 21336, 22246, 23175, 24123, 25090, 26076, 27081, 28105, 29148, 30210, 31291, 32391, 33510, 34648, 35805, 36981, 38176, 39390, 40623, 41875, 43146, 44436, 45745, 47073, 48420, 49786, 51171, 52575, 53998, 55440, 56901, 58381, 59880, 61398, 62935, 64491, 66066, 67660, 69273, 70905, 72556, 74226, 75915, 77623, 79350, 81096, 82861, 84645, 86448, 88270, 90111
Generating function in Maple syntax:
(4*x^7-4*x^6-10*x^5-6*x^4-2*x^3-2*x^2+2*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{4 x^{7}-4 x^{6}-10 x^{5}-6 x^{4}-2 x^{3}-2 x^{2}+2 x -1}{\left(x -1\right)^{3}}
Generating function in sympy syntax:
(4*x**7 - 4*x**6 - 10*x**5 - 6*x**4 - 2*x**3 - 2*x**2 + 2*x - 1)/(x - 1)**3
Implicit equation for the generating function in Maple syntax:
-(x-1)^3*F(x)+4*x^7-4*x^6-10*x^5-6*x^4-2*x^3-2*x^2+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{3} F \! \left(x \right)+4 x^{7}-4 x^{6}-10 x^{5}-6 x^{4}-2 x^{3}-2 x^{2}+2 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,n = 4,19,61-99/2*n+19/2*n^2)
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 \\ 61-\frac{99}{2} n +\frac{19}{2} n^{2} & \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) = 51
a(6) = 106
a(7) = 180
a(n) = 1/2*(-2+n)*(19*n-61), 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) = 51
a \! \left(6\right) = 106
a \! \left(7\right) = 180
a \! \left(n \right) = \frac{\left(-2+n \right) \left(19 n -61\right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16354/
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[71,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[31,x]
F[25,x] = F[19,x]+F[26,x]
F[26,x] = F[13,x]+F[14,x]+F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[19,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[16,x]*F[4,x]
F[31,x] = F[32,x]+F[41,x]
F[32,x] = F[13,x]+F[23,x]+F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[1,x]+F[19,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[29,x]+F[4,x]
F[41,x] = F[13,x]+F[42,x]+F[59,x]+F[66,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[49,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[30,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[29,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = 2*F[13,x]+F[42,x]+F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[38,x]
F[53,x] = F[54,x]
F[54,x] = F[4,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[29,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[52,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = 2*F[13,x]+F[59,x]+F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[29,x]+F[65,x]
F[65,x] = F[47,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[56,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[19,x]+F[45,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[34,x]+F[73,x]
F[73,x] = F[62,x]+F[74,x]
F[74,x] = F[13,x]+F[71,x]+F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[19,x]+F[77,x]
F[77,x] = F[13,x]+F[27,x]+F[30,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_{71}\! \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_{31}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \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_{19}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)+F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{29}\! \left(x \right)+F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right)+F_{42}\! \left(x \right)+F_{59}\! \left(x \right)+F_{66}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{30}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{29}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{42}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{38}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \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_{29}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{52}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{59}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{29}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{47}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{56}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{19}\! \left(x \right)+F_{45}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{34}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{62}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{13}\! \left(x \right)+F_{71}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{19}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{13}\! \left(x \right)+F_{27}\! \left(x \right)+F_{30}\! \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_71(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_31(x))
Eq(F_25(x), F_19(x) + F_26(x))
Eq(F_26(x), F_13(x) + F_14(x) + F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_19(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_16(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_41(x))
Eq(F_32(x), F_13(x) + F_23(x) + F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_1(x) + F_19(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_29(x) + F_4(x))
Eq(F_41(x), F_13(x) + F_42(x) + F_59(x) + F_66(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_49(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_30(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_29(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), 2*F_13(x) + F_42(x) + F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_38(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(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_29(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_52(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), 2*F_13(x) + F_59(x) + F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_29(x) + F_65(x))
Eq(F_65(x), F_47(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_56(x) + F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_19(x) + F_45(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_34(x) + F_73(x))
Eq(F_73(x), F_62(x) + F_74(x))
Eq(F_74(x), F_13(x) + F_71(x) + F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_19(x) + F_77(x))
Eq(F_77(x), F_13(x) + F_27(x) + F_30(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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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