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

1,6

Comments

For more information about the symmetric representation of sigma(n) see A237593 and A237270.

Links

Table of n, a(n) for n=1..87.

Michel Marcus, A colored version of the symmetric representation of sigma(n), multipage, n = 1..85

Example

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

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 *)

Crossrefs

Cf. A237270, A237271, A237593.

Sequence in context: A318306 A345935 A214715 * A371734 A086435 A266226

Adjacent sequences: A244142 A244143 A244144 * A244146 A244147 A244148

Keyword

nonn

Author

Michel Marcus, Jun 21 2014

Extensions

a(85) corrected by Hartmut F. W. Hoft, Jul 23 2014

Status

approved