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A229859 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of B.

%I A229859 #7 Oct 06 2013 15:29:20
%S A229859 5,8,10,15,16,20,24,25,30,32,33,35,39,40,45,48,50,51,55,56,57,60,63,
%T A229859 64,65,66,70,72,75,77,78,80,85,88,90,91,95,96,99,100,102,104,105,110,
%U A229859 112,114,115,117,120,125,126,128,130,132,135,136,140,143,144,145
%N A229859 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of B.
%C A229859 A229858 gives the values of A, and A050931 gives the values of C.
%H A229859 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer triangle</a>
%e A229859 20 appears in the sequence because there exists a 120-degree triangle with sides 12, 20 and 28.
%o A229859 (PARI)
%o A229859 \\ Gives values of B not exceeding bmax.
%o A229859 \\ e.g. t120b(40) gives [5, 8, 10, 15, 16, 20, 24, 25, 30, 32, 33, 35, 39, 40]
%o A229859 t120b(bmax) = {
%o A229859   v=pt120b(bmax);
%o A229859   s=[];
%o A229859   for(i=1, #v,
%o A229859     for(m=1, bmax\v[i],
%o A229859       if(v[i]*m<=bmax, s=concat(s, v[i]*m))
%o A229859     )
%o A229859   );
%o A229859   vecsort(s,,8)
%o A229859 }
%o A229859 \\ Gives values of B not exceeding bmax in primitive triangles.
%o A229859 \\ e.g. pt120b(40) gives [5, 8, 16, 24, 33, 35, 39]
%o A229859 pt120b(bmax) = {
%o A229859   s=[];
%o A229859   for(m=1, (bmax-1)\2,
%o A229859     for(n=1, m-1,
%o A229859       if((m-n)%3!=0 && gcd(m, n)==1,
%o A229859         a=m*m-n*n;
%o A229859         b=n*(2*m+n);
%o A229859         if(a>b, b=a);
%o A229859         if(b<=bmax, s=concat(s, b))
%o A229859       )
%o A229859     )
%o A229859   );
%o A229859   vecsort(s,,8)
%o A229859 }
%Y A229859 Cf. A050931, A229849, A229858.
%K A229859 nonn
%O A229859 1,1
%A A229859 _Colin Barker_, Oct 06 2013