Triangles and angles

[franktaw at netscape.net]

Using the PARI program:

a(n)=local(i,j,k,l); r=0; l=listcreate(n^3); for(i=1,n, for(j=1,n, 
for(k=1,n, if(gcd(i,gcd(j,k))==1&&2*max(i,max(j,k))<i+j+k, 
listput(l,(j^2+k^2-i^2)/(2*j*k)))))); listsort(l,1); matsize(Vec(l))[2]

the sequence of angle counts is:

1, 3, 9, 17, 35, 49, 82, 109, 149, 188, 262, 316, 419, 500, 607, 698, 
876, 1004, 1222, 1383, 1589, 1782, 2108, 2318, 2634, 2914, 3253, 3564, 
4088, 4411, 5000, 5392, 5917, 6410, 6995, 7468, 8308, 8926, 9661, 
10268, 11313, 11976, 13136, 13951, 14875, 15807, 17199, 18042, 19317, 
20328, 21635, 22770, 24544, 25670, 27180, 28384, 30018, 31512, 33705, 
35044

I have submitted this as A123325; I also submitted the number of 
triangles as A123324, while the number having maximum side = n is 
A123323.

Franklin T. Adams-Watters

-----Original Message-----
 From: franktaw at netscape.net [mailto:franktaw at netscape.net]

 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?

 How many different angles do those triangles have?

 Franklin T. Adams-Watters

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