A355564 Triangle read by rows: T(n,k) = n*(1+2*k) - k*(1+k), n >= 1, 0 <= k <= n-1.

1, 2, 4, 3, 7, 9, 4, 10, 14, 16, 5, 13, 19, 23, 25, 6, 16, 24, 30, 34, 36, 7, 19, 29, 37, 43, 47, 49, 8, 22, 34, 44, 52, 58, 62, 64, 9, 25, 39, 51, 61, 69, 75, 79, 81, 10, 28, 44, 58, 70, 80, 88, 94, 98, 100, 11, 31, 49, 65, 79, 91, 101, 109, 115, 119, 121

Offset: 1

Author

Lucas B. Vieira, Jul 07 2022

Keywords

• nonn
• easy
• tabl

Comments

T(n,k) is the number of potentially nonzero elements in a square, n X n band matrix of bandwidth k, i.e., the number of matrix elements (i,j) for which |i-j| <= k.

T(n,0) = n, as a zero-bandwidth matrix is diagonal, and T(n,n-1) = n^2, as the band encompasses the entire matrix.

Examples

			Triangle starts:
  1;
  2,  4;
  3,  7,  9;
  4, 10, 14, 16;
  5, 13, 19, 23, 25;
  6, 16, 24, 30, 34, 36;
  7, 19, 29, 37, 43, 47, 49;
  ...
Example: For n = 6 and k = 2, we have a band matrix of the form
    [. . . 0 0 0]
    [. . . . 0 0]
    [. . . . . 0]
    [0 . . . . .],
    [0 0 . . . .]
    [0 0 0 . . .]
where dots represent the entries which may have nonzero values. The number of such entries is T(6,2) = 24.
		

Crossrefs

The sequence is complementary to A095832, i.e.: T(n,k) = n^2 - A095832(n,k).

Differences within a row T(n,k+1) - T(n,k) give A212012.

Programs

Mathematica

Flatten[Table[n (1 + 2 k) - k (1 + k), {n, 1, 10}, {k, 0, n - 1}]]