A244145 Number of positive integers k less than n such that the symmetric representation of sigma(k) is contiguous (shares a line border) with the symmetric representation of sigma(n).
0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 3, 1, 2, 1, 2, 1, 2, 3, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 2, 3
Offset: 1
Keywords
Examples
For n = 6 the symmetric representation of sigma(6) (the outer one in the figure below) touches those for n = 4 and n = 5, so a(6) = 2. _ _ _ _ |_ _ _ |_ |_ _ _| |_ |_ _ |_ _ | |_ _|_ | | | |_ | | | | | |_|_|_|_|_|_| 1 2 3 4 5 6
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Programs
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Mathematica
(* path[n] computing the n-th Dyck path is defined in A237270 *) (* canvasNamed[] creates a matrix with the labeled symmetric regions *) (* adjacentPos[] computes list of bounding regions *) extents[n_] := Map[Transpose[#] + {{1, 0}, {1, 0}}&, Transpose[{path[n-1], Most[Rest[path[n]]]}]] squaresPos[n_] := DeleteDuplicates[Flatten[Map[Flatten[Outer[List, First[#], Last[#]], 1]&, Map[Apply[Range, #]&, extents[n], {2}]], 1]] squaresNamed[n_] := Map[#->n&, squaresPos[n]] canvasNamed[n_] := Module[{canvas = Table[0, {n}, {n}]}, ReplacePart[canvas, Flatten[Map[squaresNamed, Range[n]], 1]]] adjacentPos[n_, matrix_] := Drop[DeleteDuplicates[Flatten[Map[{matrix[[Apply[Sequence, # + {-1, 0}]]], matrix[[Apply[Sequence, # + {0, -1}]]]}&, Drop[Drop[squaresPos[n], 1], -1]], 1]], 1] a244145[n_] := Module[{c = canvasNamed[n]}, Join[{0, 1, 1}, Map[Length[adjacentPos[#, c]]&, Range[4, n]]]] a244145[87] (* computes the first 87 values *) (* Hartmut F. W. Hoft, Jul 23 2014 *)
Extensions
a(85) corrected by Hartmut F. W. Hoft, Jul 23 2014
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