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A229838 Consider all primitive 60-degree triangles with sides A < B < C. The sequence gives the values of A.

%I A229838 #20 Oct 07 2013 06:17:35
%S A229838 3,5,7,8,9,11,13,15,16,17,19,21,23,24,25,27,29,31,32,33,35,37,39,40,
%T A229838 41,43,45,47,48,49,51,53,55,56,57,59,61,63,64,65,67,69,71,72,73,75,77,
%U A229838 79,80,81,83,85,87,88,89,91,93,95,96,97,99,101,103,104,105
%N A229838 Consider all primitive 60-degree triangles with sides A < B < C. The sequence gives the values of A.
%C A229838 A primitive triangle is one for which the sides have no common factor.
%C A229838 A004611 gives the values of B, and A089025 gives the values of C.
%H A229838 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer triangle</a>
%F A229838 Empirical g.f.: -x*(x^5-x^4-x^3-2*x^2-2*x-3) / ((x-1)^2*(x^4+x^3+x^2+x+1)).
%e A229838 7 appears in the sequence because there exists a primitive 60-degree triangle with sides 7, 37 and 40.
%o A229838 (PARI)
%o A229838 \\ Gives terms not exceeding amax
%o A229838 \\ e.g. pt60a(25) gives [3,5,7,8,9,11,13,15,16,17,19,21,23,24,25]
%o A229838 pt60a(amax) = {
%o A229838   s=[];
%o A229838   for(m=1, amax\2,
%o A229838     for(n=1, m-1,
%o A229838       if((m-n)%3!=0 && gcd(m, n)==1,
%o A229838         if(2*m*n+n*n<=amax, s=concat(s, 2*m*n+n*n));
%o A229838         if(m*m-n*n<=amax, s=concat(s, m*m-n*n))
%o A229838       )
%o A229838     )
%o A229838   );
%o A229838   vecsort(s,,8)
%o A229838 }
%Y A229838 Cf. A004611, A089025, A229839.
%K A229838 nonn
%O A229838 1,1
%A A229838 _Colin Barker_, Oct 01 2013