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%I #35 Jul 23 2014 10:12:59
%S 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,
%T 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,
%U 1,3,1,3,1,2,2,2,3,2,1,2,1,2,1,4,1,2,3
%N 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).
%C For more information about the symmetric representation of sigma(n) see A237593 and A237270.
%H Michel Marcus, <a href="/A244145/a244145_3.pdf">A colored version of the symmetric representation of sigma(n), multipage, n = 1..85</a>
%e 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.
%e _ _ _ _
%e |_ _ _ |_
%e |_ _ _| |_
%e |_ _ |_ _ |
%e |_ _|_ | | |
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%e |_|_|_|_|_|_|
%e 1 2 3 4 5 6
%t (* path[n] computing the n-th Dyck path is defined in A237270 *)
%t (* canvasNamed[] creates a matrix with the labeled symmetric regions *)
%t (* adjacentPos[] computes list of bounding regions *)
%t extents[n_] := Map[Transpose[#] + {{1, 0}, {1, 0}}&, Transpose[{path[n-1], Most[Rest[path[n]]]}]]
%t squaresPos[n_] := DeleteDuplicates[Flatten[Map[Flatten[Outer[List, First[#], Last[#]], 1]&, Map[Apply[Range, #]&, extents[n], {2}]], 1]]
%t squaresNamed[n_] := Map[#->n&, squaresPos[n]]
%t canvasNamed[n_] := Module[{canvas = Table[0, {n}, {n}]}, ReplacePart[canvas, Flatten[Map[squaresNamed, Range[n]], 1]]]
%t 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]
%t a244145[n_] := Module[{c = canvasNamed[n]}, Join[{0, 1, 1}, Map[Length[adjacentPos[#, c]]&, Range[4, n]]]]
%t a244145[87] (* computes the first 87 values *)
%t (* _Hartmut F. W. Hoft_, Jul 23 2014 *)
%Y Cf. A237270, A237271, A237593.
%K nonn
%O 1,6
%A _Michel Marcus_, Jun 21 2014
%E a(85) corrected by _Hartmut F. W. Hoft_, Jul 23 2014
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