Triangles with given inradius, A057721

[all at abouthugo.de]

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?
See (German):
http://groups.google.com/group/de.sci.mathematik/msg/f9ce7bee9eafc16f

Although there are several related sequences by Reinhard Zumkeller, I could not find a sequence answering this question. Therefore I wrote a little program trying all possible side combinations up to lengths
5000 and found the following conjectured result:

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

The last column coincides with
http://www.research.att.com/~njas/sequences/A078644, that has also a comment
a(n) is the number of Pythagorean triangles with radius of the
inscribed circle equal to n. - Ant King, mathstutoring(AT)ntlworld.com,
Mar 06 2006. For number of primitive Pythagorean triangles having
inradius n, see A068068(n).

The column giving the number of all possible triangles seems to be missing in the OEIS. Can someone check the few given values and try to extend?

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
 9  12  15 3
 9  40  41 4
 9  75  78 4
10  10  12 3
10  24  26 4
10  35  39 4
11  13  20 3
11  25  30 4
11  60  61 5
11  90  97 4
11 100 109 3
12  16  20 4
12  35  37 5
12  50  58 4
12  55  65 3
12 153 159 5
13  14  15 4
13  40  51 3
13  68  75 5
14  30  40 4
15  15  24 4
15  20  25 5
15  26  37 4
15  28  41 3
15 377 388 5
16  25  39 3
17  28  39 5
17  87 100 5
18  20  34 4
18 289 305 4
19  20  37 3
19 153 170 4
21  85 104 4
25  51  74 4
27  29  52 5
27 676 701 5
28 351 377 5
31 156 185 5
33  34  65 4
36  91 125 5
39  76 113 5
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

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