A128622 Triangle T(n, k) = A128064(unsigned) * A128174, read by rows.

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

Offset

1,3

Links

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

Formula

T(n, k) = abs(A128064(n,k) * A128174(n, k), as infinite lower triangular matrices.

Sum_{k=1..n} T(n, k) = A014848(n) (row sums).

From G. C. Greubel, Mar 14 2024: (Start)

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

T(n, 1) = A109613(n+1).

T(n, n) = A000027(n).

T(2*n-1, n) = A042963(n).

T(3*n-1, n) = A016777(n+1).

T(4*n-3, n) = A047461(n).

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A319556(n).

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

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

Example

First few rows of the triangle are:
  1;
  1, 2;
  3, 2, 3;
  3, 4, 3, 4;
  5, 4, 5, 4, 5;
  5, 6, 5, 6, 5, 6;
  7, 6, 7, 6, 7, 6, 7;
  ...

Mathematica

Table[n - Mod[n+k, 2], {n, 16}, {k, n}]//Flatten (* G. C. Greubel, Mar 14 2024 *)

Prog

(Magma) [n - ((n+k) mod 2): k in [1..n], n in [1..16]]; // G. C. Greubel, Mar 14 2024
(SageMath) flatten([[n - ((n+k)%2) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 14 2024

Crossrefs

Cf. A000027, A016777, A042963, A047461, A109613, A128064, A128174.

Cf. A000326 (diagonal sums), A014848 (row sums), A319556 (alternating row sums).

Sequence in context: A304331 A241834 A086389 * A026256 A079715 A030397

Adjacent sequences: A128619 A128620 A128621 * A128623 A128624 A128625

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Mar 14 2007

Extensions

More terms added by G. C. Greubel, Mar 14 2024

Status

approved