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
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
Programs
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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)