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A357278
Perimeters of primitive integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees.
4
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.
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
KEYWORD
nonn
AUTHOR
Bernard Schott, Oct 24 2022
STATUS
approved