1,1

Descartes's theorem: 4 kissing circles with radii a,b,c,d satisfy

(1/a + 1/b + 1/c + 1/d)^2 = 2 (1/a^2 + 1/b^2 + 1/c^2 + 1/d^2).

Ray Chandler, Table of n, a(n) for n = 1..153

Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group., Discrete & Computational Geometry, 34 (2005), no. 4, 547-585.

Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings., Discrete & Computational Geometry, 35 (2006), no. 1, 1-36.

Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions., Discrete & Computational Geometry, 35 (2006), no. 1, 37-72.

Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., Apollonian Circle Packings: Number Theory, J. Number Theory, 100 (2003), 1-45.

N. J. A. Sloane, Illustration for a(1)=12. Draw 4 mutually touching circles of radii 12, 12, 3, and 1. The curvatures (1/radius) are 1/12, 1/12, 1/3, and 1, or equivalently 1, 1, 4, and 12. Four such circles can be seen (in many ways) in the illustration, which is Fig. 1 of the Graham et al. (2006, Part II) article. Used with permission of the authors.

Eric Weisstein's World of Mathematics, Soddy Circles.

Eric Weisstein's World of Mathematics, Tangent Circles

Wikipedia, Descartes' theorem

The table gives the first 8 examples:

a b c d

== == == ==

12 12 3 1

15 15 10 2

24 24 2 1

63 56 56 9

69 46 23 6

70 30 21 5

72 45 45 8

80 80 36 9

aMax = 150; (* WARNING: O(n^3) *)

Do[

If[d // IntegerQ // Not, Continue[]];

If[GCD[a, b, c, d] > 1, Continue[]];

{a, b, c, d} // Sow;

, {a, aMax}, {b, a}, {c, b},

{d, {1/(1/a + 1/b + 1/c + 2 Sqrt[(a + b + c)/(a b c)])}}

] // Reap // Last // Last // TableForm

Cf. A290931 (4 mutually tangent circles, 1 circle enclosing 3).

Sequence in context: A274550 A253235 A050480 * A063604 A357867 A350807

Adjacent sequences: A290505 A290506 A290507 * A290509 A290510 A290511

Albert Lau, Aug 04 2017

Corrected (inserted 63, 72, 234, 275) and extended by Ray Chandler, Aug 06 2017

Edited by N. J. A. Sloane, Aug 16 2017

approved