A385866 Triangle read by rows where T(n,k), for 1 <= k < n, is the row number where (n-k)^2 occurs in an n X n grid filled rowwise with the numbers 1 to n^2.

1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 5, 3, 2, 1, 1, 6, 4, 3, 2, 1, 1, 7, 5, 4, 2, 2, 1, 1, 8, 6, 4, 3, 2, 1, 1, 1, 9, 7, 5, 4, 3, 2, 1, 1, 1, 10, 8, 6, 5, 4, 3, 2, 1, 1, 1, 11, 9, 7, 6, 5, 3, 3, 2, 1, 1, 1, 12, 10, 8, 7, 5, 4, 3, 2, 2, 1, 1, 1, 13, 11, 9, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1

Offset

2,2

Comments

Rows are numbered starting from 1.

Links

Table of n, a(n) for n=2..92.

Formula

T(n, k) = floor(((n-k)^2 - 1) / n) + 1.

Example

      k=1  2  3  4  5  6
  n=2:  1
  n=3:  2, 1
  n=4:  3, 1, 1
  n=5:  4, 2, 1, 1
  n=6:  5, 3, 2, 1, 1
  n=7:  6, 4, 2, 2, 1, 1
For n = 5, the grid is:
  1  2  3  4  5
  6  7  8  9 10
 11 12 13 14 15
 16 17 18 19 20
 21 22 23 24 25
The squares (n-k)^2 = {16, 9, 4, 1} are in row numbers {4, 2, 1, 1} respectively.

Mathematica

T[n_, k_]:=Floor[((n-k)^2-1)/n]+1; Table[T[n, k], {n, 14}, {k, n-1}]//Flatten (* James C. McMahon, Jul 17 2025 *)

Crossrefs

Cf. A385865 (column position).

Sequence in context: A010766 A135841 A210992 * A220484 A174066 A089178

Adjacent sequences: A385863 A385864 A385865 * A385867 A385868 A385869

Keyword

nonn,easy,tabl

Author

Binay Krishna Maity, Jul 10 2025

Status

approved