A080251 Paired decomposition of tetrahedral numbers A000292 arranged as number triangle.

1, 2, 2, 3, 3, 4, 4, 4, 6, 6, 5, 5, 8, 8, 9, 6, 6, 10, 10, 12, 12, 7, 7, 12, 12, 15, 15, 16, 8, 8, 14, 14, 18, 18, 20, 20, 9, 9, 16, 16, 21, 21, 24, 24, 25, 10, 10, 18, 18, 24, 24, 28, 28, 30, 30, 11, 11, 20, 20, 27, 27, 32, 32, 35, 35, 36

Offset: 0

Author

Paul Barry, Feb 17 2003

Keywords

• nonn
• tabl

Comments

Used in construction of Stirling-like number triangle A080416.

Examples

			Rows are
  1;
  2, 2;
  3, 3, 4;
  4, 4, 6, 6;
  5, 5, 8, 8, 9;
  ...
Row sums are 1, 4, 10, 20, ... or C(n+3,3) = A000292(n-1).
		

Links

• R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013.
• R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.

Programs

Mathematica

T[n_, k_] := If[EvenQ[k], (k+2)(2n-k+2)/4, (k+1)(2n-k+3)/4];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 13 2018 *)
T[ n_, k_] := -(Floor(k/2) + 1)*(Floor(k/2) - n - 1); (* Michael Somos, Oct 12 2022 *)

PARI

{T(n, k) = -(k\2 + 1)*(k\2 - n - 1)}; /* Michael Somos, Oct 12 2022 */

Formula

T(n,k) = [k<=n]*floor((k+2)/2)*(n-k+floor((k+3)/2)). - Paul Barry, Jun 14 2010

Also generated by the product of pairs of integers 0 <= r1,r2 <= n whose sum is n+2.

Viewed as a square array: T(n,2*k) = k*(k+n); T(n,2*k+1) = (k+1)*(k+n). - Luc Rousseau, Dec 11 2017

Extensions

Edited by Ken Joffaniel M Gonzales, Jul 04 2010