A128177 A128174 * A004736 as infinite lower triangular matrices.

1, 2, 1, 4, 2, 1, 6, 4, 2, 1, 9, 6, 4, 2, 1, 12, 9, 6, 4, 2, 1, 16, 12, 9, 6, 4, 2, 1, 20, 16, 12, 9, 6, 4, 2, 1, 25, 20, 16, 12, 9, 6, 4, 2, 1, 30, 25, 20, 16, 12, 9, 6, 4, 2, 1, 36, 30, 25, 20, 16, 12, 9, 6, 4, 2, 1, 42, 36, 30, 25, 20, 16, 12, 9, 6, 4, 2, 1

Offset: 1

Author

Gary W. Adamson, Feb 17 2007

Keywords

• nonn
• tabl

Comments

n-th row has n nonzero terms of A002620: (1, 2, 4, 6, 9, 12, 16, ...) in reverse.

Row sums = A002623: (1, 3, 7, 13, 22, 34, 50, ...).

Examples

			First few rows of the triangle:
   1;
   2, 1;
   4, 2, 1;
   6, 4, 2, 1;
   9, 6, 4, 2, 1;
  12, 9, 6, 4, 2, 1;
  ...
		

Crossrefs

Cf. A128174, A004736, A002623, A002620.

Programs

Maple

seq(seq(floor((n-k+2)^2/4), k=1..n), n=1..20); # Ridouane Oudra, Mar 23 2024

Mathematica

T[n_,k_]:=Floor[(n-k+2)^2/4];Table[T[n,k],{n,12},{k,n}]//Flatten (* James C. McMahon, Jan 05 2025 *)

PARI

lista(nn) = {t128174 = matrix(nn, nn, n, k, (k<=n)*(1+(-1)^(n-k))/2); t004736 = matrix(nn, nn, n, k, (k<=n)*(n - k + 1)); t128177 = t128174*t004736; for (n = 1, nn, for (k = 1, n, print1(t128177[n, k], ", ");););} \\ Michel Marcus, Feb 11 2014

Formula

From Ridouane Oudra, Mar 23 2024: (Start)

T(n, k) = A002620(n-k+2), with 1 <= k <= n;

T(n, k) = floor((n-k+2)^2/4);

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

T(n, k) = (1/8)*(2*(n-k+2)^2 + (-1)^(n-k) - 1). (End)

Extensions

Partially edited and more terms from Michel Marcus, Feb 11 2014