Triangles with given inradius, A057721

[Hugo Pfoertner]

Hugo Pfoertner schrieb:
> 
> Hugo Pfoertner schrieb:
> >
> > 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?
> 
> > > 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"
> 
> An interpolating polynomial of degree 4 would be
> 
> c_min(n)=(1/12)*(n**4-2*x**3-31*x**2+152*x-60) making the next terms
> 
> 50,100,187,325,530, but that is just a guess assuming that the lower
> bound might also be a polynomial of degree 4. I'll check against my
> results for triangles with side lengths up to n=10000 tomorrow.

Forget about this silly speculation. 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.

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

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