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A264827 (a,b,c) in lexicographic order such that a^2 + b^2 + a*b - c^2 = 0 with a < b < c and gcd(a, b) = 1.

%I A264827 #21 Dec 12 2021 09:28:52
%S A264827 3,5,7,5,16,19,7,8,13,7,33,37,9,56,61,11,24,31,11,85,91,13,35,43,13,
%T A264827 120,127,15,161,169,16,39,49,17,63,73,17,208,217,19,80,91,19,261,271,
%U A264827 21,320,331,23,120,133,23,385,397,24,95,109,25,143,157
%N A264827 (a,b,c) in lexicographic order such that a^2 + b^2 + a*b - c^2 = 0 with a < b < c and gcd(a, b) = 1.
%C A264827 The sides of a primitive 120-degree integer triangle.
%H A264827 Colin Barker, <a href="/A264827/b264827.txt">Table of n, a(n) for n = 1..9999</a>
%H A264827 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_triple">Eisenstein triple</a>
%e A264827 Triples (a,b,c) begin:
%e A264827   3,  5,  7;
%e A264827   5, 16, 19;
%e A264827   7,  8, 13;
%e A264827   7, 33, 37;
%e A264827   9, 56, 61;
%e A264827   ...
%o A264827 (PARI)
%o A264827 pt120(a) = {
%o A264827   my(L=List(), n=-3*a^2, f, g, b, c);
%o A264827   fordiv(n, f,
%o A264827     g=n\f;
%o A264827     if(f>g && (g+f)%2==0 && (f-g)%4==0,
%o A264827       c=(f-g)\4; b=((f+g)\2-a)\2;
%o A264827       if(b>0 && a<b && gcd(a, b)==1, listput(L, [a,b,c]))
%o A264827     )
%o A264827   );
%o A264827   Vec(L)
%o A264827 }
%o A264827 concat(concat(vector(30, a, pt120(a))))
%Y A264827 Cf. A002476, A229849, A229858, A229859, A264826.
%K A264827 nonn,tabf
%O A264827 1,1
%A A264827 _Colin Barker_, Nov 26 2015