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

Author

Gary W. Adamson, Mar 14 2007

Keywords

• nonn
• tabl
• easy

Examples

			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;
  ...
		

Links

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

Crossrefs

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

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

Programs

Magma

[n - ((n+k) mod 2): k in [1..n], n in [1..16]]; // G. C. Greubel, Mar 14 2024

Mathematica

Table[n - Mod[n+k,2], {n,16}, {k,n}]//Flatten (* 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

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)

Extensions

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