OFFSET
1,2
COMMENTS
Suppose that R={R(n,k) : n>=1, k>=1} is a rectangular array. The accumulation array of R is given by T(n,k) = Sum_{R(i,j): 1<=i<=n, 1<=j<=k}. (See A144112.)
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 antidiagonals, flattened
FORMULA
T(n,k) = k*n*(2*n^2 + 3*(k+1)*n + 2*k^2 - 3*k + 5)/12.
EXAMPLE
The natural number array A000027 starts with
1, 2, 4, 7, ...
3, 5, 8, 12, ...
6, 9, 13, 18, ...
...
T(n,k) is the sum of numbers in the rectangle with corners at (1,1) and (n,k) of A000027, so that a corner of T is as follows:
1, 3, 7, 14, 25, 41
4, 11, 23, 42, 70, 109
10, 26, 51, 88, 140, 210
20, 50, 94, 156, 240, 350
35, 85, 155, 250, 375, 535
MATHEMATICA
f[n_, k_]:=k*n*(2n^2+3(k+1)*n+2k^2-3k+5)/12;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 29 2011
STATUS
approved