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A264826 Primitive Eisenstein triples: (a,b,c) in lexicographic order such that a^2 + b^2 - a*b - c^2 = 0, a < b < c, and gcd(a, b) = 1.

%I A264826 #12 Feb 16 2025 08:33:27
%S A264826 3,7,8,5,7,8,5,19,21,7,13,15,7,37,40,8,13,15,9,61,65,11,31,35,11,91,
%T A264826 96,13,43,48,13,127,133,15,169,176,16,19,21,16,49,55,17,73,80,17,217,
%U A264826 225,19,91,99,19,271,280,21,331,341,23,133,143,23,397,408
%N A264826 Primitive Eisenstein triples: (a,b,c) in lexicographic order such that a^2 + b^2 - a*b - c^2 = 0, a < b < c, and gcd(a, b) = 1.
%C A264826 The sides of a primitive 60-degree integer triangle.
%H A264826 Colin Barker, <a href="/A264826/b264826.txt">Table of n, a(n) for n = 1..9999</a>
%H A264826 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%H A264826 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_triple">Eisenstein triple</a>
%o A264826 (PARI)
%o A264826 pt60(a) = {
%o A264826   my(L=List(), n=-3*a^2, f, g, b, c);
%o A264826   fordiv(n, f,
%o A264826     g=n\f;
%o A264826     if(f>g && (g+f)%2==0 && (f-g)%4==0,
%o A264826       b=(f-g)\4; c=((f+g)\2+a)\2;
%o A264826       if(c>0 && a<b && gcd(a, c)==1, listput(L, [a,b,c]))
%o A264826     )
%o A264826   );
%o A264826   Vec(L)
%o A264826 }
%o A264826 concat(concat(vector(30, a, pt60(a))))
%Y A264826 Cf. A089025, A121992, A201223, A263728, A264827.
%K A264826 nonn,tabf
%O A264826 1,1
%A A264826 _Colin Barker_, Nov 26 2015