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

1,1

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.

Links

Table of n, a(n) for n=1..53.

Formula

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

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

Example

(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.

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<b then print(a+b+c); end if;
end do;
end do;

Crossrefs

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

Sequence in context: A039285 A043888 A350045 * A343067 A223442 A121594

Adjacent sequences: A357275 A357276 A357277 * A357279 A357280 A357281

Keyword

nonn

Author

Bernard Schott, Oct 24 2022

Status

approved