OFFSET
1,1
COMMENTS
The primitive areas are 168, 338, 432, 600, 768, 13608, 14280, 20280, 27216, ...
The non-primitive areas 16*a(n) are in the sequence because if R is the circumradius corresponding to a(n), then 4*R is the circumradius corresponding to 16*a(n).
Each circumradius belongs to the sequence {25, 100, 169, 225, 289, 400, 625, 676, ...}, and it seems that this last sequence is A198385 (second of a triple of squares in arithmetic progression).
The following table gives the first values (A, R, a, b, c) where A is the integer area, R the radius of the circumcircle, and a, b, c are the integer sides of the triangle.
**************************************
* A * R * a * b * c *
**************************************
* 168 * 25 * 14 * 30 * 40 *
* 336 * 25 * 14 * 48 * 50 *
* 432 * 25 * 30 * 30 * 48 *
* 600 * 25 * 30 * 40 * 50 *
* 768 * 25 * 40 * 40 * 48 *
* 2688 * 100 * 56 * 120 * 160 *
* 5376 * 100 * 56 * 192 * 200 *
* 6912 * 100 * 120 * 120 * 192 *
* 9600 * 100 * 120 * 160 * 200 *
* 12288 * 100 * 160 * 160 * 192 *
* 13608 * 225 * 126 * 270 * 360 *
* 14280 * 169 * 130 * 238 * 312 *
* 20280 * 169 * 130 * 312 * 338 *
* 27216 * 225 * 126 * 432 * 450 *
.............................
REFERENCES
Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.
LINKS
Eric W. Weisstein, MathWorld: Circumradius
FORMULA
Area A = sqrt(s*(s-a)*(s-b)*(s-c)) with s = (a+b+c)/2 (Heron's formula);
Circumradius R = a*b*c/4A.
EXAMPLE
168 is in the sequence because the area of the triangle (14, 30, 40) is given by Heron's formula A = sqrt(42*(42-14)*(42-30)*(42-40))= 168 where the number 42 is the semiperimeter, and the circumcircle is given by R = a*b*c/(4*A) = 14*30*40/(4*168) = 25, which is a square.
MATHEMATICA
nn = 1000; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 && IntegerQ[Sqrt[area2]] && IntegerQ[Sqrt[a*b*c/(4*Sqrt[area2])]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Oct 20 2013
STATUS
approved