A341969 Irregular triangle read by rows in which row n lists the sequence of widths, each contiguous sequence of identical widths w in A249223 replaced by a single entry of w, in the symmetric representation of sigma(n).

1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1

Offset: 1

Author

Hartmut F. W. Hoft, Feb 24 2021

Keywords

• nonn
• tabf

Comments

This sequence is a companion to A279387 in which each contiguous sequence of identical widths w in A249223 are replaced by a single entry of w. Using the resulting distribution pattern of widths across all parts of the symmetric representation of sigma(n) the subparts at each level are counted in A279387.

The sequence of widths are computed first to the diagonal of the symmetric representation of sigma only for those numbers in set F defined in A341971. Then the reversed list less its first number is appended so that the width at the diagonal is not listed twice. Thus every row contains an odd number of entries and is symmetric about its center entry.

Let 1 <= n, 1 <= d <= s = A001227(n) and 1 <= k <= r = floor((sqrt(8*n + 1) - 1)/2). Let Q(n,d) be row n in the triangle of A341970, R(n,d) be row n in the triangle of A341970 and S(n,d) = R(n,Q(n,d)), then T(n,e) = S(n,e) for 1 <= e <= s and T(n,e) = S(n,2*s - e) for s < e <= 2*s - 1 is row n for this sequence.

Examples

			The irregular triangle for A279387 and this sequence:
  row  A279387  A341969
  1    1        1
  2    1        1
  3    2        1  0  1
  4    1        1
  5    2        1  0  1
  6    1  1     1  2  1
  7    2        1  0  1
  8    1        1
  9    3        1  0  1  0  1
  10   2        1  0  1
  11   2        1  0  1
  12   1  1     1  2  1
  13   2        1  0  1
  14   2        1  0  1
  15   3  1     1  0  1  2  1  0  1
  16   1        1
  17   2        1  0  1
  18   1  2     1  2  1  2  1
  19   2        1  0  1
  20   1  1     1  2  1
  21   4        1  0  1  0  1  0  1
  ..   ..       ..
  30   1  3     1  2  1  2  1  2  1
  ..   ..       ..
  45   3  3     1  0  1  2  1  2  1  2  1  0  1
  ..   ..       ..
a(17)..a(21) = { 1, 0, 1, 0, 1 } is row 9; the symmetric representation of sigma(9) consists of 3 parts of width 1 - see A247687.
a(37)..a(43) = { 1, 0, 1, 2, 1, 0, 1} is row 15; the symmetric representation of sigma(15) consists of 2 outer parts of width 1 and a central part of width 2 only at the diagonal - see A338488.
a(59)..a(65) = { 1, 0, 1, 0, 1, 0, 1 } is row 21; the symmetric representation of sigma(21) consists of 4 parts of width 1, and 21 is the smallest such number - see A264102.
a(234)..a(240) = { 1, 2, 3, 2, 3, 2, 1 } is row 60; the symmetric representation of sigma(60) consists of 1 part of maximum width 3 which occurs in two subparts, and 60 is the smallest number with width 3 - see A250070.
		

Crossrefs

Cf. A235791, A237048, A237270, A237591, A237593, A247687, A249223, A249351, A250070, A264102, A279387, A280851, A338488, A341970, A341971.

Programs

Mathematica

(* function widthL[ ] is defined in A341971 *)
a341969[n_] := Module[{wL=widthL[n]}, Join[wL, Rest[Reverse[wL]]]]
Flatten[Table[a341969[n], {n, 28}]] (* the first 28 rows of the table *)

Formula

a(2*A060831(n-1) - (n-1) + e) = T(n,e), 1 <= n, 1 <= e <= 2*A001227(n) - 1.