A331222 - OEIS

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.

%I #13 Jan 13 2020 15:19:49

%S 1,16,4,81,81,3,81,256,64,256,16,25,625,625,256,625,625,25,1296,64,

%T 324,48,625,81,1296,12,625,3136,2401,2401,1225,2401,2401,1296,2401,

%U 2401,50176,4096,81,1024,49,49,4096,256,256,4096,2401,1024,4096,35721,6561

%N 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.

%C 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.

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

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

%e The first terms b(n) = a(n)/A331223(n) correspond to the following triangles (i, j, k):

%e b(1) = 1/3: (1,1,1),

%e b(2) = 16/15: (1,2,2),

%e b(3) = 4/3: (2,2,2),

%e b(4) = 81/35: (1,3,3),

%e b(5) = 81/32: (2,3,3),

%e b(6) = 3/1: (3,3,3),

%e b(7) = 81/20: (3,3,4),

%e b(8) = 256/63: (1,4,4),

%e b(9) = 64/15: (2,4,4),

%e ...

%e b(41) = b(42) = 49/3: (5,7,8), (7,7,7).

%Y Cf. A317182, A331223, A331227, A331228.

%K nonn,frac

%O 1,2

%A _Hugo Pfoertner_, Jan 12 2020