Jaap Spies j.spies at hccnet.nl
Mon Mar 24 23:24:24 CET 2003

Should this be added to the comment of A000005



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

See 'Opgave A'


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.


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.

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