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A265189 Soddy circles: the two circles tangent to each of three mutually tangent circles.

%I A265189 #23 Feb 16 2025 08:33:27
%S A265189 69,46,23,6,138,70,30,21,5,105,132,33,11,4,-132,138,92,46,12,276,140,
%T A265189 60,42,10,210,153,136,72,17,306,207,138,69,18,414,210,90,63,15,315,
%U A265189 216,135,24,10,-135,238,119,102,21,357,252,63,28,9,0
%N A265189 Soddy circles: the two circles tangent to each of three mutually tangent circles.
%C A265189 For any three mutually tangent circles (with radii a, b, and c), one can construct a fourth circle (the inner Soddy circle, with radius d) that is mutually tangent internally to the three circles, and a fifth circle (the outer Soddy circle, with radius e) that is mutually tangent externally to the three circles. For this sequence all five radii have integral lengths.
%C A265189 The sequence is an array of 5-tuples (a,b,c,d,e) ordered by increasing values of a, with a > b > c.
%C A265189 A positive value for the outer Soddy circle indicates that it contains the three circles; a negative value indicates that it is exterior to the three circles; a value of 0 indicates that it has an infinite radius, that is, it is a straight line.
%H A265189 Colin Barker, <a href="/A265189/b265189.txt">Table of n, a(n) for n = 1..1000</a>
%H A265189 Kival Ngaokrajang, <a href="/A265189/a265189.pdf">Illustration of a(1) - a(5), a(41) - a(45) and a(51) - a(55)</a>
%H A265189 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SoddyCircles.html">Soddy Circles</a>
%H A265189 Wikipedia, <a href="http://en.wikipedia.org/wiki/Descartes%27_theorem">Descartes' theorem</a>
%o A265189 (PARI)
%o A265189 soddy(amax) = {
%o A265189   my(L=List(), abc, t, u);
%o A265189   for(a=1, amax,
%o A265189     for(b=1, a-1,
%o A265189       for(c=1, b-1,
%o A265189         abc=a*b*c;
%o A265189         if(issquare(abc*(a+b+c), &t),
%o A265189           u=a*b+a*c+b*c;
%o A265189           if(abc%(u+2*t) == 0,
%o A265189             if(u-2*t != 0,
%o A265189               if(abc%(u-2*t) == 0,
%o A265189                 listput(L, [a,b,c,abc\(u+2*t),-abc\(u-2*t)])
%o A265189               )
%o A265189             ,
%o A265189               listput(L, [a,b,c,abc\(u+2*t),0])
%o A265189             )
%o A265189           )
%o A265189         )
%o A265189       )
%o A265189     )
%o A265189   );
%o A265189   Vec(L)
%o A265189 }
%o A265189 soddy(253)
%Y A265189 Cf. A256694.
%Y A265189 See also the many sequences arising from Apollonian circle packing: A135849, A137246, A154636, etc.
%Y A265189 Also the sequences related to Soddy's circle packings: A046159, A046160, A062536, etc.
%K A265189 sign,tabf
%O A265189 1,1
%A A265189 _Colin Barker_, Dec 04 2015