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

1,1

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.

Links

Table of n, a(n) for n=1..63.

Example

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

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<b then print(b); end if;
end do;
end do;

Crossrefs

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

Cf. also A088586, A229849, A229859.

Sequence in context: A242090 A030006 A229849 * A088586 A350347 A073136

Adjacent sequences: A357273 A357274 A357275 * A357277 A357278 A357279

Keyword

nonn

Author

Bernard Schott, Sep 25 2022

Status

approved