A210484 Ordered areas of primitive integer Soddyian triangles.

12, 252, 1872, 8400, 17100, 27900, 75852, 178752, 191100, 261072, 378432, 705600, 737100, 1063692, 1343100, 1976400, 2317392, 3483900, 3820752, 4038012, 6061692, 6760512, 8822352, 9305100, 9909900, 12024012

Offset: 1

Author

Frank M Jackson, Jan 23 2013

Keywords

• nonn

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.

Examples

			a(3)=1872 given by m=3, n=1
		

Links

• Nikolaos Dergiades, The Soddy circles, Forum Geom., 7 (2007) 191-197.
• Frank M. Jackson, Soddyian triangles, Forum Geom., 13 (2013) 1-6.

Crossrefs

Subsequence of A367737.

Programs

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}]]]

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).