0123_0231_1032_2031_2310
Counting sequence:
1, 1, 2, 6, 19, 51, 107, 189, 297, 431, 591, 777, 989, 1227, 1491, 1781, 2097, 2439, 2807, 3201, 3621, 4067, 4539, 5037, 5561, 6111, 6687, 7289, 7917, 8571, 9251, 9957, 10689, 11447, 12231, 13041, 13877, 14739, 15627, 16541, 17481, 18447, 19439, 20457, 21501, 22571, 23667, 24789, 25937, 27111, 28311, 29537, 30789, 32067, 33371, 34701, 36057, 37439, 38847, 40281, 41741, 43227, 44739, 46277, 47841, 49431, 51047, 52689, 54357, 56051, 57771, 59517, 61289, 63087, 64911, 66761, 68637, 70539, 72467, 74421, 76401, 78407, 80439, 82497, 84581, 86691, 88827, 90989, 93177, 95391, 97631, 99897, 102189, 104507, 106851, 109221, 111617, 114039, 116487, 118961, 121461
Generating function in Maple syntax:
-(2*x^7+5*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{2 x^{7}+5 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:
(-2*x**7 - 5*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)+2*x^7+5*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)+2 x^{7}+5 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 = 4,19,13*n^2-87*n+161)
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 \\ 13 n^{2}-87 n +161 & \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) = 107
a(7) = 189
a(n) = 13*n^2-87*n+161, 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) = 107
a \! \left(7\right) = 189
a \! \left(n \right) = 13 n^{2}-87 n +161, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17261/
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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[4,x]
F[18,x] = F[15,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[59,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[29,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[1,x]+F[4,x]
F[28,x] = F[15,x]
F[29,x] = F[30,x]+F[43,x]
F[30,x] = F[21,x]+F[22,x]+F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[1,x]+F[25,x]
F[36,x] = F[21,x]+F[37,x]+F[42,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[4,x]
F[42,x] = F[33,x]*F[4,x]
F[43,x] = 2*F[21,x]+F[44,x]+F[52,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[48,x]
F[46,x] = F[47,x]
F[47,x] = F[40,x]
F[48,x] = F[49,x]
F[49,x] = 2*F[21,x]+F[44,x]+F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[36,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[36,x]
F[55,x] = F[56,x]
F[56,x] = 2*F[21,x]+F[52,x]+F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[39,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[64,x]
F[61,x] = F[33,x]+F[62,x]
F[62,x] = F[21,x]+F[37,x]+F[63,x]
F[63,x] = F[4,x]*F[61,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[21,x]+F[59,x]+F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[25,x]+F[47,x]
F[68,x] = 2*F[21,x]+F[52,x]+F[57,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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{59}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{43}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \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_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{36}\! \left(x \right) = F_{21}\! \left(x \right)+F_{37}\! \left(x \right)+F_{42}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \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_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{44}\! \left(x \right)+F_{52}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{40}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{44}\! \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_{36}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{36}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{52}\! \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_{39}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{64}\! \left(x \right)
F_{61}\! \left(x \right) = F_{33}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{21}\! \left(x \right)+F_{37}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{21}\! \left(x \right)+F_{59}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{25}\! \left(x \right)+F_{47}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{52}\! \left(x \right)+F_{57}\! \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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_15(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_59(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_29(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_1(x) + F_4(x))
Eq(F_28(x), F_15(x))
Eq(F_29(x), F_30(x) + F_43(x))
Eq(F_30(x), F_21(x) + F_22(x) + F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_1(x) + F_25(x))
Eq(F_36(x), F_21(x) + F_37(x) + F_42(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_4(x))
Eq(F_42(x), F_33(x)*F_4(x))
Eq(F_43(x), 2*F_21(x) + F_44(x) + F_52(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_48(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_40(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), 2*F_21(x) + F_44(x) + F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_36(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_36(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), 2*F_21(x) + F_52(x) + F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_39(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_64(x))
Eq(F_61(x), F_33(x) + F_62(x))
Eq(F_62(x), F_21(x) + F_37(x) + F_63(x))
Eq(F_63(x), F_4(x)*F_61(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_21(x) + F_59(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_25(x) + F_47(x))
Eq(F_68(x), 2*F_21(x) + F_52(x) + F_57(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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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