0123_0132_0321_1023_2310
Counting sequence:
1, 1, 2, 6, 19, 49, 93, 142, 193, 248, 307, 370, 437, 508, 583, 662, 745, 832, 923, 1018, 1117, 1220, 1327, 1438, 1553, 1672, 1795, 1922, 2053, 2188, 2327, 2470, 2617, 2768, 2923, 3082, 3245, 3412, 3583, 3758, 3937, 4120, 4307, 4498, 4693, 4892, 5095, 5302, 5513, 5728, 5947, 6170, 6397, 6628, 6863, 7102, 7345, 7592, 7843, 8098, 8357, 8620, 8887, 9158, 9433, 9712, 9995, 10282, 10573, 10868, 11167, 11470, 11777, 12088, 12403, 12722, 13045, 13372, 13703, 14038, 14377, 14720, 15067, 15418, 15773, 16132, 16495, 16862, 17233, 17608, 17987, 18370, 18757, 19148, 19543, 19942, 20345, 20752, 21163, 21578, 21997
Generating function in Maple syntax:
-(2*x^9-3*x^8-9*x^7-3*x^6+8*x^5+6*x^4+2*x^3+2*x^2-2*x+1)/(x-1)^3
Generating function in latex syntax:
-\frac{2 x^{9}-3 x^{8}-9 x^{7}-3 x^{6}+8 x^{5}+6 x^{4}+2 x^{3}+2 x^{2}-2 x +1}{\left(x -1\right)^{3}}
Generating function in sympy syntax:
(-2*x**9 + 3*x**8 + 9*x**7 + 3*x**6 - 8*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)+2*x^9-3*x^8-9*x^7-3*x^6+8*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)+2 x^{9}-3 x^{8}-9 x^{7}-3 x^{6}+8 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,n = 5,49,n = 6,93,2*n^2+21*n-103)
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 \\ 49 & n =5 \\ 93 & n =6 \\ 2 n^{2}+21 n -103 & \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) = 49
a(6) = 93
a(7) = 142
a(8) = 193
a(9) = 248
a(n) = 2*n^2+21*n-103, n >= 10
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) = 49
a \! \left(6\right) = 93
a \! \left(7\right) = 142
a \! \left(8\right) = 193
a \! \left(9\right) = 248
a \! \left(n \right) = 2 n^{2}+21 n -103, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16218/
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[19,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[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[76,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[33,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[4,x]
F[26,x] = F[15,x]+F[27,x]+F[30,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[17,x]+F[29,x]
F[29,x] = F[14,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[12,x]+F[32,x]
F[32,x] = F[16,x]
F[33,x] = F[34,x]+F[44,x]
F[34,x] = F[15,x]+F[21,x]+F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[1,x]+F[12,x]
F[40,x] = F[15,x]+F[41,x]+F[43,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[14,x]+F[4,x]
F[43,x] = F[37,x]*F[4,x]
F[44,x] = F[15,x]+F[45,x]+F[60,x]+F[74,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[54,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[4,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[32,x]
F[54,x] = F[55,x]
F[55,x] = 2*F[15,x]+F[45,x]+F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[32,x]+F[4,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[40,x]
F[63,x] = F[64,x]
F[64,x] = F[15,x]+F[65,x]+F[71,x]+F[72,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[69,x]
F[67,x] = F[68,x]
F[68,x] = x^2
F[69,x] = F[70,x]
F[70,x] = F[32,x]*F[4,x]
F[71,x] = F[4,x]*F[62,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[58,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[73,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[36,x]+F[78,x]
F[78,x] = F[64,x]+F[79,x]
F[79,x] = F[15,x]+F[41,x]+F[80,x]
F[80,x] = F[36,x]*F[4,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_{19}\! \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_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{76}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right)+F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{44}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \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_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{14}\! \left(x \right)+F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{60}\! \left(x \right)+F_{74}\! \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_{54}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \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_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{32}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{32}\! \left(x \right)+F_{4}\! \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_{40}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \left(x \right)+F_{65}\! \left(x \right)+F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{69}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = x^{2}
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{58}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{36}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{64}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \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_19(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_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_76(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_33(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_4(x))
Eq(F_26(x), F_15(x) + F_27(x) + F_30(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_17(x) + F_29(x))
Eq(F_29(x), F_14(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_12(x) + F_32(x))
Eq(F_32(x), F_16(x))
Eq(F_33(x), F_34(x) + F_44(x))
Eq(F_34(x), F_15(x) + F_21(x) + F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_1(x) + F_12(x))
Eq(F_40(x), F_15(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_14(x) + F_4(x))
Eq(F_43(x), F_37(x)*F_4(x))
Eq(F_44(x), F_15(x) + F_45(x) + F_60(x) + F_74(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_54(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_4(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_32(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), 2*F_15(x) + F_45(x) + F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_32(x) + F_4(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_40(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_15(x) + F_65(x) + F_71(x) + F_72(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_69(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), x**2)
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_32(x)*F_4(x))
Eq(F_71(x), F_4(x)*F_62(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_58(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_73(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_36(x) + F_78(x))
Eq(F_78(x), F_64(x) + F_79(x))
Eq(F_79(x), F_15(x) + F_41(x) + F_80(x))
Eq(F_80(x), F_36(x)*F_4(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:
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