A000005

[Jaap Spies]

Should this be added to the comment of A000005

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000005

?

a(n) is the number of Pythagorean triangles with radius of the inscribed circle equal n,
n = 1, 2, 3, ...

See 'Opgave A'

http://www.math.leidenuniv.nl/~naw/serie5/deel04/mrt2003/pdf/problemen-uwc.pdf

If you do not read Dutch: Give all Pythagorean triangles ... (see above).


Jaap Spies


If you want to solve this problem on your own, do not look below.

Spoiler:




















Let P(u,v)=(a,b,c) be the parameterization of the Pythagorean triangle
with a^2 + b^2 = c^2 and
a = u^2 - v^2, b = 2uv and c = u^2 + v^2 (u > v).

The radius of the inscribed circle is $r = \frac {2A} {a+b+c}$
(A is area of the triangle)
So r = ... = v(u-v) and therefor we have for each divisor v of n = r
an integer u = n/v + v with P(u,v) as corresponding Pythagorean triangle.
So the number of solutions for radius n is the number of divisors of n:
tau(n) or sigma_0(n), see A000005.