OFFSET
1,2
COMMENTS
Row 3 is = 3rd triangular number + 3rd square + 3rd pentagonal number + 3rd hexagonal number + ... + 3rd k-gonal number. First column is triangular numbers. A086271 Rectangular array T(n,k) of polygonal numbers, by diagonals.
LINKS
Eric Weisstein's World of Mathematics, Polygonal Number. See equation (4), our partial sums are on this as array element values.
FORMULA
a(k,n) = (k*(k-1)/2)n^2 + (k*(k+3)/4)n. a(k,n) = row k of array of partial sums = k-th triangular number + k-th square + k-th pentagonal number + k-th hexagonal number + ... = A000217(k) + A000290(k) + A000326(k) + A000384(k) + ... a(1,n) = n. a(2,n) = (n(n+1)/2)-3 = A000217(n) - 3. a(3,n) = 3*n(n+3)/2 = A000096 with offset 3.
EXAMPLE
Partial row sum array begins:
1 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... n.
2 | 3, 7, 12, 18, 25, 33, 42, 52, (n(n+1)/2)-3.
3 | 6, 15, 27, 42, 60, 81, 105, ... (3/2)n^2 + (9/2) n.
4 | 10, 26, 48, 76, 110, 150, ... 3n^2 + 7n.
5 | 15, 40, 75, ... 5n^2 + 10n.
6 | 21, 57, 108, ... (15/2)n^2 + (27/2)n.
MAPLE
A086271 := proc(n, k) k*binomial(n, 2)+n ; end: A125764 := proc(n, k) add(A086271(n, i), i=1..k) ; end: for d from 1 to 15 do for k from 1 to d do printf("%d, ", A125764(d-k+1, k)) ; od: od: # R. J. Mathar, Nov 02 2007
CROSSREFS
KEYWORD
AUTHOR
Jonathan Vos Post and Joshua Zucker, Feb 03 2007
EXTENSIONS
More terms from R. J. Mathar, Nov 02 2007
Keyword tabl added by Michel Marcus, Apr 08 2013
STATUS
approved