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%I #22 Dec 11 2015 08:12:26
%S 69,46,23,6,138,70,30,21,5,105,132,33,11,4,-132,138,92,46,12,276,140,
%T 60,42,10,210,153,136,72,17,306,207,138,69,18,414,210,90,63,15,315,
%U 216,135,24,10,-135,238,119,102,21,357,252,63,28,9,0
%N Soddy circles: the two circles tangent to each of three mutually tangent circles.
%C 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 The sequence is an array of 5-tuples (a,b,c,d,e) ordered by increasing values of a, with a > b > c.
%C 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 Colin Barker, <a href="/A265189/b265189.txt">Table of n, a(n) for n = 1..1000</a>
%H Kival Ngaokrajang, <a href="/A265189/a265189.pdf">Illustration of a(1) - a(5), a(41) - a(45) and a(51) - a(55)</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SoddyCircles.html">Soddy Circles</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Descartes%27_theorem">Descartes' theorem</a>
%o (PARI)
%o soddy(amax) = {
%o my(L=List(), abc, t, u);
%o for(a=1, amax,
%o for(b=1, a-1,
%o for(c=1, b-1,
%o abc=a*b*c;
%o if(issquare(abc*(a+b+c), &t),
%o u=a*b+a*c+b*c;
%o if(abc%(u+2*t) == 0,
%o if(u-2*t != 0,
%o if(abc%(u-2*t) == 0,
%o listput(L, [a,b,c,abc\(u+2*t),-abc\(u-2*t)])
%o )
%o ,
%o listput(L, [a,b,c,abc\(u+2*t),0])
%o )
%o )
%o )
%o )
%o )
%o );
%o Vec(L)
%o }
%o soddy(253)
%Y Cf. A256694.
%Y See also the many sequences arising from Apollonian circle packing: A135849, A137246, A154636, etc.
%Y Also the sequences related to Soddy's circle packings: A046159, A046160, A062536, etc.
%K sign,tabf
%O 1,1
%A _Colin Barker_, Dec 04 2015
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