A128621 A127648 * A128174 as an infinite lower triangular matrix.

1, 0, 2, 3, 0, 3, 0, 4, 0, 4, 5, 0, 5, 0, 5, 0, 6, 0, 6, 0, 6, 7, 0, 7, 0, 7, 0, 7, 0, 8, 0, 8, 0, 8, 0, 8, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 11, 0, 11, 0, 11, 0, 11, 0, 11, 0, 11, 0, 12, 0, 12, 0, 12, 0, 12, 0, 12, 0, 12, 13, 0, 13, 0, 13, 0, 13, 0, 13, 0, 13, 0, 13

Offset: 1

Author

Gary W. Adamson, Mar 14 2007

Keywords

• nonn
• easy
• tabl

Examples

			First few rows of the triangle:
  1;
  0, 2;
  3, 0, 3;
  0, 4, 0, 4;
  5, 0, 5, 0, 5;
  ...
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Crossrefs

Cf. A000326, A123684, A127648, A128174.

Cf. A093005 (row sums).

Programs

Magma

[n*(1+(-1)^(n+k))/2: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 13 2024

Mathematica

Table[n*(1+(-1)^(n+k))/2, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 13 2024 *)

SageMath

flatten([[n*(1+(-1)^(n+k))//2 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 13 2024

Formula

Odd rows: n terms of n, 0, n, ...; even rows, n terms of 0, n, 0, ...

T(n,k) = n if n+k even, T(n,k) = 0 if n+k odd.

Sum_{k=1..n} T(n, k) = A093005(n) (row sums).

From G. C. Greubel, Mar 13 2024: (Start)

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

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n+1)*A093005(n).

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n) * A000326(floor((n+1)/2)).

Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1 - (-1)^n)*A123684(floor((n+1)/2)). (End)

Extensions

More terms added by G. C. Greubel, Mar 13 2024