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

1,1

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

Table of n, a(n) for n=1..26.

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

Subsequence of A367737.

Sequence in context: A245917 A265453 A367737 * A099139 A289565 A198475

Adjacent sequences: A210481 A210482 A210483 * A210485 A210486 A210487

Keyword

nonn

Author

Frank M Jackson, Jan 23 2013

Status

approved