A357276 Middle side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3 = 120 degrees.

5, 8, 16, 24, 33, 35, 39, 56, 45, 63, 51, 85, 80, 57, 77, 95, 120, 120, 88, 91, 115, 143, 112, 161, 105, 175, 165, 195, 208, 160, 168, 145, 224, 203, 187, 221, 155, 261, 217, 192, 279, 209, 288, 247, 320, 272, 323, 280, 231, 315, 273, 259, 385, 357, 333, 304, 399, 352, 253, 407, 299, 287, 440

Offset: 1

Author

Bernard Schott, Sep 25 2022

Keywords

• nonn

Comments

The triples of sides (a,b,c) with a < b < c are in nondecreasing order of largest side c, and if largest sides coincide, then by increasing order of the smallest side. This sequence lists the b's.

For the corresponding primitive triples and miscellaneous properties and references, see A357274.

Solutions b of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.

Also, b is generated by integers u, v such that gcd(u,v) = 1 and 0 < v < u, with b = 2*u*v + v^2.

This sequence is not increasing. For example, a(8) = 56 for triangle with largest side c = 61 while a(9) = 45 for triangle with largest side c = 67.

Differs from A088586, the first 20 terms are the same then a(21) = 115 while A088586(21) = 143.

A229849 gives all the possible values of the middle side b, in increasing order without repetition, for primitive triples, while A229859 gives all the possible values of the middle side b, in increasing order without repetition, but for all triples, not necessarily primitive.

Examples

			a(17) = a(18) = 120 since 17th and 18th triples are respectively (13, 120, 127) and (23, 120, 133).
		

Crossrefs

Cf. A357274 (triples), A357275 (smallest side), this sequence (middle side), A357277 (largest side), A357278 (perimeter).

Cf. also A088586, A229849, A229859.

Programs

Maple

for c from 5 to 500 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a,b)=1 and a