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

0,3

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.

Links

Table of n, a(n) for n=0..59.

Formula

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

Example

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
========================================================================

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

Crossrefs

Cf. A000027, A000096, A006002, A048395.

Sequence in context: A085009 A192883 A011435 * A264299 A243445 A227569

Adjacent sequences: A139306 A139307 A139308 * A139310 A139311 A139312

Keyword

easy,nonn,tabl

Author

Jonathan Vos Post, Jun 07 2008

Extensions

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

Status

approved