Triangles with given inradius, A057721

[Graeme McRae]

There are 12 primitive triangles with inradius n=3, when you define 
primitive as GCD(a,b,c,n)=1.  That's sequence A120252.

----- Original Message ----- 
From: "David Wilson" <davidwwilson at comcast.net>
To: "seqfan" <seqfan at ext.jussieu.fr>
Sent: Tuesday, June 13, 2006 7:50 PM
Subject: Re: Triangles with given inradius, A057721


> T(r) = set of integer triangles with inradius r, so that { |T(n)| } = 
> A120062.
> P(r) = set of primitive integer triangles with inradius r.
>
> Let sequences t = { |T(n)| } = A120062, p = { |P(n)| }
>
> You suggest that T(r) = UNION(d | r, r/d x P(d)), by which I mean that 
> T(r) consists of the primitive triangles in P(d) magnified by a factor of 
> r/d, for each d dividing r.  The union is disjoint, which would imply
>
> [1]  t(r) = SUM(d | r, p(d))
>
> Unfortunately, this is not quite true.  By [1], we would expect
>
> [2]  t(3) = p(1) + p(3).
>
> Now, there is only one primitive triangle with inradius 1, namely (3,4,5), 
> so p(1) = 1, and there are ten primitive triangles of inradius 3,
>
> (7,24,25) (7,65,68) (8,15,17) (11,13,20) (12,55,65)
> (13,40,51) (15,28,41) (16,25,39) (19,20,37) (11,100,109)
>