A059036 In a triangle of numbers (such as that in A059032, A059033, A059034) how many entries lie above position (n,k)? Answer: T(n,k) = (n+1)*(k+1)-1 (n >= 0, k >= 0).

0, 1, 1, 2, 3, 2, 3, 5, 5, 3, 4, 7, 8, 7, 4, 5, 9, 11, 11, 9, 5, 6, 11, 14, 15, 14, 11, 6, 7, 13, 17, 19, 19, 17, 13, 7, 8, 15, 20, 23, 24, 23, 20, 15, 8, 9, 17, 23, 27, 29, 29, 27, 23, 17, 9, 10, 19, 26, 31, 34, 35, 34, 31, 26, 19, 10, 11, 21, 29, 35, 39, 41

Offset: 0

Author

N. J. A. Sloane, Feb 13 2001

Keywords

• nonn
• tabl
• easy

Examples

			As an infinite triangular array:
  0
  1   1
  2   3   2
  3   5   5   3
  4   7   8   7   4
  5   9  11  11   9   5
As an infinite square array (matrix):
  0   1   2   3   4   5
  1   3   5   7   9  11
  2   5   8  11  14  17
  3   7  11  15  19  23
  4   9  14  19  24  29
  5  11  17  23  29  35
		

Crossrefs

T(n, k) = A003991(n, k) - 1.

Cf. A000005, A005581.

Programs

PARI

{T(n, k) = n + k + n*k}; /* Michael Somos, Jul 28 2015 */

Formula

T(n, k) = max(T(n-1, k-1), T(n-1, k)) + min(k, n-k+1). - Jon Perry, Aug 05 2004

E.g.f.: exp(x+y)(x+y+xy) (as a square array read by antidiagonals). - Paul Barry, Sep 24 2004

From Michael Somos, Jul 28 2015: (Start)

Row sums = Sum_{k=0..n} T(n-k, k) = A005581(n+1).

T(n, k) = T(k, n) = T(-2-n, -2-k) for all n, k in Z.

Sum_{n, k >= 0} x^T(n, k) = f(x) / x where f() is the g.f. for A000005. (End)