A274550 The curvature of smallest circle among 4 mutually tangent(externally) circles with integer curvature and primitive (share no common factor).

12, 15, 23, 24, 28, 33, 34, 35, 38, 39, 40, 42, 45, 47, 50, 52, 53, 56, 57, 58, 59, 60, 61, 62, 63, 63, 64, 66, 69, 71, 72, 72, 73, 76, 77, 77, 79, 80, 81, 82, 82, 83, 84, 84, 85, 86, 87, 87, 88, 90, 91, 91, 94, 94, 95, 95, 96, 96, 97, 98, 98, 99, 99

Offset

1,1

Comments

4 mutually tangent circles satisfy 2 (a^2 + b^2 + c^2 + d^2) = (a + b + c + d)^2 where a,b,c,d are the curvatures.

Links

Table of n, a(n) for n=1..63.

Wikipedia, Apollonian gasket

Example

a,  b,  c, d
12, 4,  1, 1
15, 3,  2, 2
23, 6,  3, 2
24, 12, 1, 1
28, 9,  4, 1

Mathematica

aMax = 100;
Do[
    If[GCD[a, b, c] > 1, Continue[]];
    d = a + b + c - 2 Sqrt[a b + a c + b c];
    If[d // IntegerQ // Not, Continue[]];
    (*{a, b, c, d}*)a // Sow;
    , {a, aMax}
    , {b, (2 a)/Sqrt[3] - a // Ceiling, (Sqrt[a] - 1)^2}
    , {c, (a-b)^2/(4(a+b))//Ceiling, Min[b, (Sqrt[a]-Sqrt[b])^2-1//Ceiling]}
] // Reap // Last // Last(*//TableForm*)
d =.;

Crossrefs

Sequence in context: A158190 A122040 A342758 * A253235 A050480 A290508

Adjacent sequences: A274547 A274548 A274549 * A274551 A274552 A274553

Keyword

nonn

Author

Albert Lau, Jul 03 2016

Status

approved