A331227 a(n) = numerator of squared radius R^2 of the circumcircle of the n-th triangle with integer sides i <= j <= k in a list of such triangles, the list being sorted by increasing size of R. Denominators are A331228.

1, 16, 4, 16, 81, 81, 3, 81, 256, 64, 64, 256, 16, 25, 625, 625, 256, 625, 1600, 81, 625, 25, 2025, 1296, 64, 64, 324, 48, 625, 81, 400, 1296, 5184, 12, 625, 3136, 2401, 3969, 2401, 1225, 1225, 2401, 2401, 1296, 2401, 784, 2401, 50176, 6400, 4096, 81, 1024, 49, 49, 49, 49

Offset: 1

Author

Hugo Pfoertner, Jan 12 2020

Keywords

• nonn
• frac

Comments

Radii shared by more than one triangle are not removed. The first occurrence is for squared radius 64/15 at positions n = 10 and n = 11.

Examples

			The first terms b(n) = a(n)/A331228(n) correspond to the following triangles (i, j, k):
  b(1) = 1/3: (1,1,1),
  b(2) = 16/15: (1,2,2),
  b(3) = 4/3: (2,2,2),
  b(4) = 16/7: (2,2,3) (obtuse triangle excluded in A331222),
  b(5) = 81/35: (1,3,3),
  b(6) = 81/32: (2,3,3),
  b(7) = 3/1: (3,3,3),
  b(8) = 81/20: (3,3,4),
  b(9) = 256/63: (1,4,4),
  b(10) = 64/15: (2,3,4), (obtuse)
  b(11) = 64/15: (2,4,4).
		

Crossrefs

Cf. A331222, A331223, A331228.

Formula

Squared radius of circumcircle of triangle with sides a, b, c:

R^2 = (a*b*c)^2 / (16*s*(s - a)*(s - b)*(s - c)) with s = (a + b + c)/2.