OFFSET
1,2
COMMENTS
Triangle-tree numbers: a(n) = sum(b(m), m = 1..n), b(m) = 1,1,2,1,2,3,1,2,3,4,... = A002260. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
REFERENCES
W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge, 1993, p. 159.
FORMULA
a(n*(n+1)/2+m)=n*(n+1)*(n+2)/6 + m*(m+1)/2=A000292(n)+ A000217(m), m=0...n+1, n=1, 2, 3.. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
a(n) = a(n-1)+A002260(n). As a triangle with n >= k >= 1: a(n, k) =a(n-1, k)+(n-1)*n/2 =a(n, k-1)+k =(n^3-n+3k^2+3k)/6. - Henry Bottomley, Nov 14 2001
a(n) = b(n) * (b(n) + 1) * (b(n) + 2) / 6 + c(n) * (c(n) + 1) / 2, where b(n) = [sqrt(2 * n) - 1/2] and c(n) = n - b(n) * (b(n) + 1) / 2 - Robert A. Stump (bee_ess107(AT)msn.com), Sep 20 2002
As a triangle, T(n,k) = binomial(n+1, 3) + binomial(k+1,2). - Franklin T. Adams-Watters, Jan 27 2014
EXAMPLE
The triangle starts:
1
2 4
5 7 10
11 13 16 20
21 23 26 30 35
MAPLE
a := 0: for i from 1 to 15 do for j from 1 to i do a := a+j: printf(`%d, `, a); od:od:
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Feb 05 2000
STATUS
approved