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A210484 Ordered areas of primitive integer Soddyian triangles.

%I A210484 #24 Dec 09 2023 13:48:15
%S A210484 12,252,1872,8400,17100,27900,75852,178752,191100,261072,378432,
%T A210484 705600,737100,1063692,1343100,1976400,2317392,3483900,3820752,
%U A210484 4038012,6061692,6760512,8822352,9305100,9909900,12024012
%N A210484 Ordered areas of primitive integer Soddyian triangles.
%C A210484 A Soddyian triangle is a triangle whose outer Soddy circle has degenerated into a straight line. If it is assumed that the sides a<=b<=c then, 1/Sqrt(s-c) = 1/Sqrt(s-a)+1/Sqrt(s-b) where s is the semiperimeter. All integer Soddyian triangles are Heronian. It is conjectured that a(n) has no multiplicities - checked to a(21886129).
%C A210484 a(n) == 0 mod 12.
%H A210484 Nikolaos Dergiades, <a href="http://forumgeom.fau.edu/FG2007volume7/FG200726index.html">The Soddy circles</a>, Forum Geom., 7 (2007) 191-197.
%H A210484 Frank M. Jackson, <a href="http://forumgeom.fau.edu/FG2013volume13/FG201301index.html">Soddyian triangles</a>, Forum Geom., 13 (2013) 1-6.
%F A210484 Areas generated by m, n coprime with m >= n as area = m^2*n^2*(m+n)^2*(m^2+m*n+n^2).
%e A210484 a(3)=1872 given by m=3, n=1
%t A210484 getpairs[k_] := (list = IntegerPartitions[k, {2}]; n = 1; acceptlist = {}; While[n <= Length[list], If[GCD[list[[n]][[1]], list[[n]][[2]]]==1, (acceptlist=Append[acceptlist, n]; n++), n++]]; Reverse[Table[list[[n]], {n, acceptlist}]]);
%t A210484 getlist[j_] := (newlist = getpairs[j]; Table[newlist[[m]][[1]]^2*newlist[[m]][[2]]^2(newlist[[m]][[1]]+newlist[[m]][[2]])^2(newlist[[m]][[1]]^2+newlist[[m]][[2]]^2+newlist[[m]][[1]]*newlist[[m]][[2]]), {m,1,Length[newlist]}]);
%t A210484 maxLen = 15; Sort[Flatten[Table[getlist[p], {p,2,maxLen}]]]
%Y A210484 Subsequence of A367737.
%K A210484 nonn
%O A210484 1,1
%A A210484 _Frank M Jackson_, Jan 23 2013