OFFSET
1,1
COMMENTS
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).
a(n) == 0 mod 12.
LINKS
Nikolaos Dergiades, The Soddy circles, Forum Geom., 7 (2007) 191-197.
Frank M. Jackson, Soddyian triangles, Forum Geom., 13 (2013) 1-6.
FORMULA
Areas generated by m, n coprime with m >= n as area = m^2*n^2*(m+n)^2*(m^2+m*n+n^2).
EXAMPLE
a(3)=1872 given by m=3, n=1
MATHEMATICA
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}]]);
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]}]);
maxLen = 15; Sort[Flatten[Table[getlist[p], {p, 2, maxLen}]]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, Jan 23 2013
STATUS
approved