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

2,5

Comments

Columns are numbered starting from 1.

Links

Table of n, a(n) for n=2..93.

Formula

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

Example

      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.

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 *)

Prog

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

Crossrefs

Cf. A385866.

Sequence in context: A299537 A053239 A046569 * A046596 A174093 A204028

Adjacent sequences: A385862 A385863 A385864 * A385866 A385867 A385868

Keyword

nonn,easy,tabl

Author

Binay Krishna Maity, Jul 10 2025

Status

approved