Triangles with given inradius, A057721

[Hugo Pfoertner]

Graeme McRae schrieb:
> 
> Mathworld has an article on Heronian Triangles in which a parameterized
> formula is given that gives all rational-sided triangles with rational area.
> Using integer values for the parameters, and scaling down all triangles that
> are not "primitive" (in this sense, primitive means no common divisor among
> the sides or inradius), I was able to quickly generate all the primitive
> triangles with inradii 1, 2, 3, ... up to 12.
> 
> The number of primitive integer-sided triangles for each radius is 1, 4, 12,
> 13, 14, 28, 23, 27, 38, 32, 25, 81.  Can someone check that?
> 
> Then, assuming that's right, the total number of triangles is the sum of the
> primitive triangles for the given radius plus those of all the radii that
> are factors of the given radius.  So, for example, when the inradius is 12,
> the number of triangles is 81+1+4+12+13+28=139 -- that is, the 81 primitive
> triangles plus the primitive triangles for each of 12's factors 1, 2, 3, 4,
> and 6.
> 
> Using this method, the sequence of integer-sided triangles with inradius n
> is 1, 5, 13, 18, 15, 45, 24, 45, 51, 51, 26, 139.  Can someone check that,
> too?
> 
> Then, after someone has verified these numbers, I will go ahead and extend
> A120062, and add the reference to mathworld, along with the method of
> calculating the sequence.  I will also do a superseeker, and then submit
> another sequence (and/or comment an existing sequence) that contains just
> the primitive triangles, and cross-reference them to each other.  Thanks
> very much!

I just want to point those who can read German to Thomas Mautsch's
contribution in the German maths newsgroup de.sci.mathematik.

Thomas has extended everything up to 100 terms and has given a lot of
useful explanations:
http://groups.google.com/group/de.sci.mathematik/msg/88541089a50e7b68

Hugo