Triangles with given inradius, A057721

[Hugo Pfoertner]

all at abouthugo.de schrieb:
> 
> SeqFans,
> 
> on Jun 9 a question related to incircles of triangles was raised in
the German maths newsgroup de.sci.mathematik.
> Ulrich Diez asked: Can we find all triangles with integer sides that
have a given integer inradius n?
>
[...]
Inradius
|   Number of all possible triangles
|   |  Pythagorean triangles 
1   1  1
2   5  2
3  13  3
4  18  3
5  15  3
6  45  6
[...]

Thanks to Jud McCranie for checking up to triangle side lengths 2000;
his results confirm the correctness of 1,5,13,18,15,... and also that
the longest side for inradius 6 is A057721(6)=1405.

I've now submitted http://www.research.att.com/~njas/sequences/A120062
 
> Looking at the list of found triangles with sides a<=b<c, e.g. for
Inradius<=5:
> 
>  a   b   c Inradius
>  3   4   5 1
>  5  12  13 2
>  6   8  10 2
>  6  25  29 2
>  7  15  20 2
>  7  24  25 3
>  7  65  68 3
>  8  15  17 3
>  8  26  30 3
>  9  10  17 2
[...]
> 51  52 101 5
> 
> and checking the minimum and maximum lengths of the longest triangle
side c for a given inradius n one finds
> 
> n cmin  cmax
> 1   5     5
> 2  10    29
> 3  12   109
> 4  15   305
> 5  25   701
>

Most amazingly there seems to be no sequence containing
5,10,12,15,25 in the OEIS.

So this is a candidate for a sequence
"Minimum possible longest side of a triangle with integer sides and
inradius n"
 
> http://www.research.att.com/~njas/sequences/A057721
> n^4 + 3*n^2 + 1. Author: njas
> seems to be the upper bound for the longest side. Can someone try to
explain?
> 
> Hugo Pfoertner