A062536 Increasing values for the radius of the inner Soddy circle associated with three unequal kissing circles, the four radii of the system forming a primitive quadruple.

5, 9, 17, 36, 39, 64, 74, 81, 100

Offset: 1

Author

Lekraj Beedassy, Jun 25 2001

Keywords

• more
• nonn

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).

Examples

			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.
		

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.