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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.
15
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
1,2
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.
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)/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
KEYWORD
nonn,frac
AUTHOR
Hugo Pfoertner, Jan 12 2020
STATUS
approved