A139309 Array by antidiagonals, sum of non-k-gonal numbers between consecutive k-gonal numbers.

0, 0, 2, 0, 5, 9, 0, 9, 26, 24, 0, 14, 51, 75, 50, 0, 20, 84, 153, 164, 90, 0, 27, 125, 258, 342, 305, 147, 0, 35, 174, 390, 584, 645, 510, 224, 0, 44, 231, 549, 890, 1110, 1089, 791, 324, 0, 54, 296, 735, 1260, 1700, 1884, 1701, 1160, 450, 0, 65, 369, 948, 1694

Offset: 0

Author

Jonathan Vos Post, Jun 07 2008

Keywords

• easy
• nonn
• tabl

Comments

The n=1 column is A000096(k) = n*(n+3)/2. The k=3 row is the sum of nontriangular numbers between successive triangular numbers (A006002) = the sum of n consecutive integers beginning with (n-th triangular number)+1 = (n*(n+1)^2)/2. The k=4 row is the sum of nonsquares between successive squares (A048395) = 2*n^3 + 2*n^2 + n. The k=5 row is the sum of non-pentagonal numbers between successive pentagonal numbers. The k-th row is the sum of non-k-gonal numbers between successive k-gonal numbers. Each column is a quadratic sequence. Each row is a cubic sequence.

Examples

			The array begins:
========================================================================
...|.n=0.|.n=1.|.n=2.|.n=3.|.n=4.|.n=5.|.n=6.|.n=7.|.n=8.|.n=9.|.in.OEIS
====|=====|=====|=====|=====|=====|=====|=====|=====|=====|=====|========
k=3.|..0..|..2..|..9..|..24.|..50.|..90.|.147.|.224.|.324.|.450.|.A006002
k=4.|..0..|..5..|.26..|..75.|.164.|.305.|.510.|.791.|1160.|1629.|.A048395
k=5.|..0..|..9..|.51..|.153.|.342.|.645.|1089.|...................not.yet
k=6.|..0..|.14..|.84..|.258.|.584.|...............................not.yet
k=7.|..0..|.20..|125..|.390.|.....................................not.yet
k=8.|..0..|.27..|174..|...........................................not.yet
k=9.|..0..|.35..|231..|...........................................not.yet
k=10|..0..|.44..|296..|...........................................not.yet
========================================================================
		

Crossrefs

Cf. A000027, A000096, A006002, A048395.

Programs

Maple

A139309 := proc(k,n) n*(k-2)*((k-2)*n^2+1+2*n)/2 ; end: for d from 3 to 16 do for n from 0 to d-3 do printf("%d,", A139309(d-n,n)) ; od: od: # R. J. Mathar, Jun 12 2008

Formula

T(k,n) = n(k-2)((k-2)n^2+1+2n)/2. - R. J. Mathar, Jun 12 2008

Extensions

More terms from R. J. Mathar, Jun 12 2008