A385865 Triangle read by rows where T(n,k), for 1 <= k < n, is the column number where (n-k)^2 occurs in an n X n grid filled rowwise with the numbers 1 to n^2.

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

Offset: 2

Author

Binay Krishna Maity, Jul 10 2025

Keywords

• nonn
• easy
• tabl

Comments

Columns are numbered starting from 1.

Examples

			      k=1  2  3  4  5  6
  n=2:  1
  n=3:  1, 1
  n=4:  1, 4, 1
  n=5:  1, 4, 4, 1
  n=6:  1, 4, 3, 4, 1
  n=7:  1, 4, 2, 2, 4, 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 column numbers {1, 4, 4, 1} respectively.
		

Crossrefs

Cf. A385866.

Programs

Mathematica

T[n_,k_]:=Mod[k^2-1,n]+1;Table[T[n,k],{n,0,14},{k,n-1}]//Flatten (* James C. McMahon, Jul 16 2025 *)

PARI

row(n) = vector(n-1, k, (k^2-1) % n + 1); \\ Michel Marcus, Jul 11 2025

Formula

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