OFFSET
1,1
COMMENTS
A family of nonsquare values for a(n) may be generated by the formula a(n) = n{(n + 2)^2 + 1 }/2 for n not a multiple of 5. For some values of a(n) which are squares, the three kissing circles share a common external tangent and their radii are related by 1/sqrt(x) = 1/sqrt(y) + 1/sqrt(z).
LINKS
Pat Ballew, Soddy's Formula
Thesaurus.maths.org, Soddy's Formula or Descartes' Circle Theorem
Eric Weisstein's World of Mathematics, Soddy Circles.
FORMULA
The inner Soddy circle radius r is explicitly given by 1/r = 1/x + 1/y + 1/z + 2/R with R^2 = xyz/(x + y +z) where x, y, z are the kissing circles' radii and R the radius of the circle orthogonal to the latter three.
EXAMPLE
The quadruples (9,28,63,252) and (74,312,481,888) for instance are respectively the 2nd and 7th primitive solution set (r,x,y,z) satisfying the given explicit formula for r.
CROSSREFS
KEYWORD
more,nonn
AUTHOR
Lekraj Beedassy, Jun 25 2001
STATUS
approved