A125764 Array of partial sums of rows of array in A086271, read by antidiagonals.

1, 3, 2, 6, 7, 3, 10, 15, 12, 4, 15, 26, 27, 18, 5, 21, 40, 48, 42, 25, 6, 28, 57, 75, 76, 60, 33, 7, 36, 77, 108, 120, 110, 81, 42, 8, 45, 100, 147, 174, 175, 150, 105, 52, 9, 55, 126, 192, 238, 255, 240, 196, 132, 63, 10, 66, 155, 243, 312, 350, 351, 315, 248, 162, 75, 11

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

Table of n, a(n) for n=1..66.

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

Cf. A000096, A000217, A000290, A000326, A000384, A086270, A086271, A086272, A086273.

Sequence in context: A233208 A196518 A207636 * A023897 A267100 A178754

Adjacent sequences: A125761 A125762 A125763 * A125765 A125766 A125767

Keyword

easy,nonn,tabl

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