A331222 a(n) = numerator of squared radius R^2 of the circumcircle of the n-th non-obtuse triangle with integer sides i <= j <= k <= sqrt(i^2 + j^2) in a list of such triangles, the list being sorted by increasing size of R. Denominators are A331223.

1, 16, 4, 81, 81, 3, 81, 256, 64, 256, 16, 25, 625, 625, 256, 625, 625, 25, 1296, 64, 324, 48, 625, 81, 1296, 12, 625, 3136, 2401, 2401, 1225, 2401, 2401, 1296, 2401, 2401, 50176, 4096, 81, 1024, 49, 49, 4096, 256, 256, 4096, 2401, 1024, 4096, 35721, 6561

Offset

1,2

Comments

Radii shared by more than one triangle are not removed. The first occurrence is for squared radius 49/3 at positions n = 41 and n = 42.

Links

Table of n, a(n) for n=1..51.

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.

Example

The first terms b(n) = a(n)/A331223(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) = 81/35: (1,3,3),
  b(5) = 81/32: (2,3,3),
  b(6) = 3/1: (3,3,3),
  b(7) = 81/20: (3,3,4),
  b(8) = 256/63: (1,4,4),
  b(9) = 64/15: (2,4,4),
...
  b(41) = b(42) = 49/3: (5,7,8), (7,7,7).

Crossrefs

Cf. A317182, A331223, A331227, A331228.

Sequence in context: A331227 A040247 A245826 * A089083 A363191 A102411

Adjacent sequences: A331219 A331220 A331221 * A331223 A331224 A331225

Keyword

nonn,frac

Author

Hugo Pfoertner, Jan 12 2020

Status

approved