Voozh

A185506

Accumulation array, T, of the natural number array A000027, by antidiagonals.

1, 3, 4, 7, 11, 10, 14, 23, 26, 20, 25, 42, 51, 50, 35, 41, 70, 88, 94, 85, 56, 63, 109, 140, 156, 155, 133, 84, 92, 161, 210, 240, 250, 237, 196, 120, 129, 228, 301, 350, 375, 374, 343, 276, 165, 175, 312, 416, 490, 535, 550, 532, 476, 375, 220, 231, 415, 558, 664, 735, 771, 770, 728, 639, 495, 286

(list; table; graph; refs; listen; history; text; internal format)

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.)

The formula for the integer T(n,k) has denominator 12. The 2nd, 3rd, and 4th accumulation arrays of A000027 have formulas in which the denominators are 144, 2880, and 86400, respectively; see A185507, A185508, and A185509.

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