A350045 -id:A350045 - cp's OEIS frontend

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

Bernard Schott, Oct 24 2022

nonn

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.

			(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.

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

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).

897^2 + 560^2 + 897*560 = 1273^2, 168^2 + 1235^2 + 168*1235 = 1327^2 and 897 + 560 + 1273 = 168 + 1235 + 1327 = 2730. So 2730 is a term. 38640^2 + 5291^2 + 38640*5291 = 41539^2, 23088^2 + 22715^2 + 23088*22715 = 39667^2, 10857^2 + 34040^2 + 10857*34040 = 40573^2 and 38640 + 5291 + 41539 = 23088 + 22715 + 39667 = 10857 + 34040 + 40573 = 85470. So 85470 is a term.

def A(n) ary = [] (1..n).each{|i| (i + 1..n).each{|j| if i.gcd(j) == 1 && (i - j) % 3 > 0 ary << 2 * j * j + 3 * i * j + i * i end } } ary end p A(200).group_by(&:to_i).select{|k, v| v.size > 1}.keys.sort[0..50]

cp's OEIS Frontend

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

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A350047 Perimeters of more than one primitive 120-degree integer triangle.