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 A265189 Soddy circles: the two circles tangent to each of three mutually tangent circles. 3
 69, 46, 23, 6, 138, 70, 30, 21, 5, 105, 132, 33, 11, 4, -132, 138, 92, 46, 12, 276, 140, 60, 42, 10, 210, 153, 136, 72, 17, 306, 207, 138, 69, 18, 414, 210, 90, 63, 15, 315, 216, 135, 24, 10, -135, 238, 119, 102, 21, 357, 252, 63, 28, 9, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. The sequence is an array of 5-tuples (a,b,c,d,e) ordered by increasing values of a, with a > b > 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. LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Kival Ngaokrajang, Illustration of a(1) - a(5), a(41) - a(45) and a(51) - a(55) Eric Weisstein's World of Mathematics, Soddy Circles Wikipedia, Descartes' theorem PROG (PARI) soddy(amax) = {   my(L=List(), abc, t, u);   for(a=1, amax,     for(b=1, a-1,       for(c=1, b-1,         abc=a*b*c;         if(issquare(abc*(a+b+c), &t),           u=a*b+a*c+b*c;           if(abc%(u+2*t) == 0,             if(u-2*t != 0,               if(abc%(u-2*t) == 0,                 listput(L, [a, b, c, abc\(u+2*t), -abc\(u-2*t)])               )             ,               listput(L, [a, b, c, abc\(u+2*t), 0])             )           )         )       )     )   );   Vec(L) } soddy(253) CROSSREFS Cf. A256694. See also the many sequences arising from Apollonian circle packing: A135849, A137246, A154636, etc. Also the sequences related to Soddy's circle packings: A046159, A046160, A062536, etc. Sequence in context: A292606 A036181 A033389 * A345486 A253430 A256694 Adjacent sequences:  A265186 A265187 A265188 * A265190 A265191 A265192 KEYWORD sign,tabf AUTHOR Colin Barker, Dec 04 2015 STATUS approved

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Last modified July 24 01:23 EDT 2021. Contains 346269 sequences. (Running on oeis4.)