Triangles with given inradius, A057721

[Hugo Pfoertner]

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.

> 
> > 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?

See Joseph Myers' post.

> >
> > Hugo Pfoertner