Triangles and angles

[Henry]

Perhaps:
Start with b(n) = floor[(n+1)^2/4]: 1,2,4,6,9,12,16,20,25,30,36,42,... 
(number of triangles, integer edges, positive area, longest edge n, i.e. 
A002620 offset).
Let c(n) = b(n) - sum_k c(k)  where  k<n and k divides n: 
1,1,3,4,8,7,15,14,21,20,35,26,...
e.g. c(12) = b(12)-c(6)-c(4)-c(3)-c(2)-c(1) = 42-7-4-3-1-1 = 26 (to 
remove similar triangles).
Take sums d(n) = d(n-1) + c(n): 1,2,5,9,17,24,39,53,74,94,129,155,... 
(to get longest edge<=n).

So for example d(4)=9 corresponding to (1,1,1), (1,2,2), (2,2,3), 
(1,3,3), (2,3,3), (2,3,4), (3,3,4), (1,4,4), (3,4,4)

Henry Bottomley

franktaw at netscape.net wrote:

> Consider all triangles with integral sides, no side longer than n.
> How many different triangles are there, up to similarity?  I.e., how 
> many triples a,b,c with a<=b<=c<a+b, c <= n, and gcd(a,b,c) = 1?