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A274550 The curvature of smallest circle among 4 mutually tangent(externally) circles with integer curvature and primitive (share no common factor). 0
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 (list; graph; refs; listen; history; text; internal format)
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: A259040 A158190 A122040 * A253235 A050480 A290508

Adjacent sequences:  A274547 A274548 A274549 * A274551 A274552 A274553

KEYWORD

nonn

AUTHOR

Albert Lau, Jul 03 2016

STATUS

approved

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Last modified March 7 19:13 EST 2021. Contains 341928 sequences. (Running on oeis4.)