A139309 - OEIS

%I #9 Oct 03 2015 23:17:52

%S 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,

%T 342,305,147,0,35,174,390,584,645,510,224,0,44,231,549,890,1110,1089,

%U 791,324,0,54,296,735,1260,1700,1884,1701,1160,450,0,65,369,948,1694

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

%C 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.

%F T(k,n) = n(k-2)((k-2)n^2+1+2n)/2. - _R. J. Mathar_, Jun 12 2008

%e The array begins:

%e ========================================================================

%e ...|.n=0.|.n=1.|.n=2.|.n=3.|.n=4.|.n=5.|.n=6.|.n=7.|.n=8.|.n=9.|.in.OEIS

%e ====|=====|=====|=====|=====|=====|=====|=====|=====|=====|=====|========

%e k=3.|..0..|..2..|..9..|..24.|..50.|..90.|.147.|.224.|.324.|.450.|.A006002

%e k=4.|..0..|..5..|.26..|..75.|.164.|.305.|.510.|.791.|1160.|1629.|.A048395

%e k=5.|..0..|..9..|.51..|.153.|.342.|.645.|1089.|...................not.yet

%e k=6.|..0..|.14..|.84..|.258.|.584.|...............................not.yet

%e k=7.|..0..|.20..|125..|.390.|.....................................not.yet

%e k=8.|..0..|.27..|174..|...........................................not.yet

%e k=9.|..0..|.35..|231..|...........................................not.yet

%e k=10|..0..|.44..|296..|...........................................not.yet

%e ========================================================================

%p 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

%Y Cf. A000027, A000096, A006002, A048395.

%K easy,nonn,tabl

%O 0,3

%A _Jonathan Vos Post_, Jun 07 2008

%E More terms from _R. J. Mathar_, Jun 12 2008