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%I #17 Oct 29 2022 10:41:55
%S 15,28,40,66,77,91,104,126,144,153,170,187,190,209,220,228,260,276,
%T 286,299,322,325,350,345,390,400,420,435,442,464,476,493,496,522,527,
%U 544,558,551,589,608,620,646,630,665,672,714,703,740,777,770,798,814,805
%N Perimeters of primitive integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees.
%C This sequence lists the sums a+b+c of the triples of sides (a,b,c) of A357274.
%C 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.
%C For miscellaneous properties, links and references, see A357274.
%C 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.
%C Perimeters are in increasing order without repetition in A350045 and perimeters that appear more than once are in A350047.
%F a(n) = A357274(n, 1) + A357274(n, 2) + A357274(n, 3).
%F a(n) = A357275(n) + A357276(n) + A357277(n).
%e (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.
%p for c from 5 to 100 by 2 dofor a from 3 to c-2 do
%p b := (-a + sqrt(4*c^2-3*a^2))/2;
%p if b=floor(b) and gcd(a,b)=1 and a<b then print(a+b+c); end if;
%p end do;
%p end do;
%Y Cf. A350045 (perimeters without repetition), A350047, A357274 (triples), A357275 (smallest side), A357276 (middle side), A357277 (largest side).
%K nonn
%O 1,1
%A _Bernard Schott_, Oct 24 2022
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