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

1,3

Links

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

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)

Example

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

Mathematica

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

Prog

(Magma) [n*(1+(-1)^(n+k))/2: k in [1..n], n in [1..15]]; // 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

Crossrefs

Cf. A000326, A123684, A127648, A128174.

Cf. A093005 (row sums).

Sequence in context: A195673 A339674 A241070 * A132385 A358840 A191716

Adjacent sequences: A128618 A128619 A128620 * A128622 A128623 A128624

Keyword

nonn,easy,tabl

Author

Gary W. Adamson, Mar 14 2007

Extensions

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

Status

approved