A357278 Perimeters of primitive integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees.

15, 28, 40, 66, 77, 91, 104, 126, 144, 153, 170, 187, 190, 209, 220, 228, 260, 276, 286, 299, 322, 325, 350, 345, 390, 400, 420, 435, 442, 464, 476, 493, 496, 522, 527, 544, 558, 551, 589, 608, 620, 646, 630, 665, 672, 714, 703, 740, 777, 770, 798, 814, 805

Offset: 1

Author

Bernard Schott, Oct 24 2022

Keywords

• nonn

Comments

This sequence lists the sums a+b+c of the triples of sides (a,b,c) of A357274.

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

For miscellaneous properties, links and references, see A357274.

This sequence is not increasing. For example, a(23) = 350 for triangle with longest side = 163 while a(24) = 345 for triangle with longest side = 169.

Perimeters are in increasing order without repetition in A350045 and perimeters that appear more than once are in A350047.

Examples

			(3, 5, 7) is the smallest triple in A357274 with 7^2 = 3^2 + 3*5 + 5^2, so a(1) = 3 + 5 + 7 = 15.
		

Crossrefs

Cf. A350045 (perimeters without repetition), A350047, A357274 (triples), A357275 (smallest side), A357276 (middle side), A357277 (largest side).

Programs

Maple

for c from 5 to 100 by 2 dofor 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
				

Formula

a(n) = A357274(n, 1) + A357274(n, 2) + A357274(n, 3).

a(n) = A357275(n) + A357276(n) + A357277(n).