A290931 Radius of a circle enclosing three mutually tangent circles, such that they have coprime integer radii and with collectively six distinct points of tangency.

6, 6, 15, 18, 20, 21, 28, 35, 40, 40, 42, 42, 45, 45, 52, 54, 56, 63, 66, 66, 72, 75, 77, 88, 91, 95, 99, 100, 104, 105, 105, 110, 112, 117, 120, 126, 130, 143, 153, 153, 156, 160, 160, 165, 165, 168, 170, 175, 186, 187, 189, 190, 195, 196, 198, 198, 204, 208, 208

Offset

1,1

Comments

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).

When the largest circle encloses other 3 circles, its radius is negative.

If all circles are tangent to each other at the same point, Descartes's theorem does not apply. In this case, all circles can have any radius.

Links

Ray Chandler, Table of n, a(n) for n = 1..837

Eric Weisstein's World of Mathematics, Soddy Circles.

Eric Weisstein's World of Mathematics, Tangent Circles

Wikipedia, Descartes' theorem

Example

The table gives the first 8 examples:
   a  b  c  d
  == == == ==
   6  3  2  1
   6  3  3  2
  15 10  3  2
  18  9  8  8
  20 12  5  3
  21 14  7  6
  28 21  4  3
  35 15 14  6

Mathematica

aMax = 150; (* WARNING: O(n^3) *)
Do[
     If[x \[NotElement] Rationals, Continue[]];
     {d1, d2} = 1/(-(1/a) + 1/b + 1/c + {1, -1} 2 x);
     If[GCD[a, b, c, d1] == 1, {a, b, c, d1} // Sow];
     If[d2 > c || d2 == d1, Continue[]];
     If[GCD[a, b, c, d2] == 1, {a, b, c, d2} // Sow];
     , {a, aMax}, {b, 2, a - 2}, {c, Min[b, a - b]}
     , {x, {Sqrt[(-a + b + c)/(-a b c)]}}] // Reap // Last // Last // TableForm

Crossrefs

Cf. A290508 (4 circles tangent externally).

Sequence in context: A266223 A256675 A362534 * A257372 A058563 A175561

Adjacent sequences: A290928 A290929 A290930 * A290932 A290933 A290934

Keyword

nonn

Author

Albert Lau, Aug 13 2017

Extensions

Description clarified by Ray Chandler, Aug 19 2017

Status

approved