0123_1032_2301_2310
Counting sequence:
1, 1, 2, 6, 20, 56, 114, 194, 296, 420, 566, 734, 924, 1136, 1370, 1626, 1904, 2204, 2526, 2870, 3236, 3624, 4034, 4466, 4920, 5396, 5894, 6414, 6956, 7520, 8106, 8714, 9344, 9996, 10670, 11366, 12084, 12824, 13586, 14370, 15176, 16004, 16854, 17726, 18620, 19536, 20474, 21434, 22416, 23420, 24446, 25494, 26564, 27656, 28770, 29906, 31064, 32244, 33446, 34670, 35916, 37184, 38474, 39786, 41120, 42476, 43854, 45254, 46676, 48120, 49586, 51074, 52584, 54116, 55670, 57246, 58844, 60464, 62106, 63770, 65456, 67164, 68894, 70646, 72420, 74216, 76034, 77874, 79736, 81620, 83526, 85454, 87404, 89376, 91370, 93386, 95424, 97484, 99566, 101670, 103796
Generating function in Maple syntax:
-(12*x^5+7*x^4+2*x^3+2*x^2-2*x+1)/(x-1)^3
Generating function in latex syntax:
-\frac{12 x^{5}+7 x^{4}+2 x^{3}+2 x^{2}-2 x +1}{\left(x -1\right)^{3}}
Generating function in sympy syntax:
(-12*x**5 - 7*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)+12*x^5+7*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)+12 x^{5}+7 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 = 2,2,11*n^2-63*n+96)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 11 n^{2}-63 n +96 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 56
a(n) = 11*n^2-63*n+96, n >= 6
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) = 20
a \! \left(5\right) = 56
a \! \left(n \right) = 11 n^{2}-63 n +96, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15169/
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[27,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[16,x]+F[23,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]+F[4,x]
F[19,x] = F[20,x]
F[20,x] = x^2
F[21,x] = F[22,x]
F[22,x] = F[15,x]+F[16,x]+F[20,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[4,x]
F[25,x] = F[15,x]+F[23,x]+F[26,x]
F[26,x] = x^2
F[27,x] = F[2,x]+F[28,x]
F[28,x] = F[15,x]+F[29,x]+F[62,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[39,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[1,x]+F[4,x]
F[35,x] = F[15,x]+F[16,x]+F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[4,x]
F[38,x] = F[15,x]+F[20,x]+F[26,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[15,x]+F[29,x]+F[36,x]
F[41,x] = F[15,x]+F[42,x]+F[53,x]+F[61,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[51,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[4,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[38,x]
F[51,x] = F[52,x]
F[52,x] = 2*F[15,x]+F[42,x]+F[49,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[38,x]
F[56,x] = F[57,x]
F[57,x] = F[15,x]+F[53,x]+F[58,x]+F[60,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[46,x]
F[60,x] = 0
F[61,x] = 0
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[24,x]+F[64,x]
F[64,x] = F[65,x]+F[67,x]
F[65,x] = F[15,x]+F[62,x]+F[66,x]
F[66,x] = F[32,x]*F[4,x]
F[67,x] = F[15,x]+F[53,x]+F[58,x]+F[68,x]
F[68,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_{27}\! \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_{16}\! \left(x \right)+F_{23}\! \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_{18}\! \left(x \right)+F_{21}\! \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) = x^{2}
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \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_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = x^{2}
F_{27}\! \left(x \right) = F_{2}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{62}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{39}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \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_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \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_{38}\! \left(x \right)+F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{26}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{53}\! \left(x \right)+F_{61}\! \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_{51}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \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_{38}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{49}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{38}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{46}\! \left(x \right)
F_{60}\! \left(x \right) = 0
F_{61}\! \left(x \right) = 0
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{24}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{67}\! \left(x \right) = F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \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_27(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) + F_16(x) + F_23(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x) + F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), x**2)
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_4(x))
Eq(F_25(x), F_15(x) + F_23(x) + F_26(x))
Eq(F_26(x), x**2)
Eq(F_27(x), F_2(x) + F_28(x))
Eq(F_28(x), F_15(x) + F_29(x) + F_62(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_39(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_1(x) + F_4(x))
Eq(F_35(x), F_15(x) + F_16(x) + F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_4(x))
Eq(F_38(x), F_15(x) + F_20(x) + F_26(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_15(x) + F_29(x) + F_36(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_53(x) + F_61(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_51(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_4(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_38(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), 2*F_15(x) + F_42(x) + F_49(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_38(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_15(x) + F_53(x) + F_58(x) + F_60(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_46(x))
Eq(F_60(x), 0)
Eq(F_61(x), 0)
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_24(x) + F_64(x))
Eq(F_64(x), F_65(x) + F_67(x))
Eq(F_65(x), F_15(x) + F_62(x) + F_66(x))
Eq(F_66(x), F_32(x)*F_4(x))
Eq(F_67(x), F_15(x) + F_53(x) + F_58(x) + F_68(x))
Eq(F_68(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": [1, 0, 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]]}], "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": [1, 0, 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]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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