Triangles with given inradius, A057721

[PFOERTNER, Hugo]

-----Original Message-----
From: Hugo Pfoertner [mailto:all at abouthugo.de] 
Sent: Monday, June 12, 2006 22:01
To: seqfan at ext.jussieu.fr
Subject: Re: Triangles with given inradius, A057721


Hugo Pfoertner wrote:
> 
I now have results up to
inradius=10, which needs to extend the search for the longest side until
10^4+3*10^2+1=10301. As can be seen in the table below searching for the
shortest side can stop much earlier. The table contains the number of
triangles and the minimal and maximal side lengths a<=b<c among all
integer sided triangles with given inradius.


Table extended to n<=15
-----------------------

Rho  d  amin  amax bmin    bmax  cmin      cmax
(n)
 1   1     3     3    4       4     5         5
 2   5     5     9    8      25    10        29
 3  13     7    19   10     100    12       109
 4  18     9    33   14     289    15       305
 5  15    11    51   20     676    25       701
 6  45    13    73   20    1369    24      1405
 7  24    15    99   28    2500    35      2549
 8  45    17   129   28    4225    30      4289
 9  51    19   163   30    6724    36      6805
10  52    21   201   39   10201    39     10301
11  26    23   243   44   14884    55     15005
12 139    25   289   40   21025    45     21169
13  31    27   339   52   28900    65     29069
14  80    29   393   56   38809    63     39005
15 110    31   451   50   51076    53     51301

The additional results for n=11...15 agree with the results found by
Thomas Mautsch; I'll submit the two new sequences for bmin and cmin as
A120063 and A120064 this evening.

d is A120062, amin is 2*n+1; amax seems to be 2*n^2+1 (OEIS A058331)
aus. bmin is new and needs to be explained. bmax apparently is
n^4+2*n^2+1 (A082044). cmin is new and needs an explanation. cmax is
n^4+3*n^2+1 (A057721), as proved by Joseph Myers.

Hugo Pfoertner