Triangles and angles

[franktaw at netscape.net]

It's related, but not on the money.  A015616 doesn't constrain the 
triples to be a triangle, so it allows triples like 1,2,3 and 2,3,7.  
It also has strict inequality for the three terms, so it doesn't allow 
things like 1,2,2 and 2,2,3, which I do want to allow.

I believe the number of triangles starts:

1,2,5,9,17,24,39

- the first few triangles are 1,1,1; 1,2,2; 1,3,3; 2,2,3; 2,3,3; 1,4,4; 
2,3,4; 3,3,4; 3,4,4 -

and the number of angles starts:

1,3,9,17,35,49,82

If you enumerate the triangles, you can count the angles using the law 
of cosines: cos A = (b^2 + c^2 - a^2) / (2bc).

Franklin T. Adams-Watters


-----Original Message-----
From: aplewe at sbcglobal.net

A015616? The description reads " Number of triples (i,j,k) with 1 <= 
i<j<k
<= n and GCD{i,j,k} = 1." Below are a.) the intermediate terms (which
appears to be the same as A000741), and b.) the running total, which is 
the
same as A015616, based on my hand calculations.

3 = 1 (1)
4 = 3 (4)
5 = 6 (10)
6 = 9 (19)
7 = 15 (34)
8 = 18 (52)

    -Andrew Plewe-


-----Original Message-----
From: franktaw at netscape.net [mailto:franktaw at netscape.net]
Sent: Monday, September 25, 2006 2:29 PM
To: seqfan at ext.jussieu.fr
Subject: Triangles and angles

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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