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A229858 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of A.

%I A229858 #19 Jun 30 2023 22:42:42
%S A229858 3,5,6,7,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,
%T A229858 29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,
%U A229858 52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70
%N A229858 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of A.
%C A229858 A229859 gives the values of B, and A050931 gives the values of C.
%C A229858 This sequence contains every integer larger than 8. - _Nathaniel Johnston_, Oct 06 2013
%H A229858 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer triangle</a>
%H A229858 Kival Ngaokrajang, <a href="/A229858/a229858.pdf">Illustration of initial terms</a>
%H A229858 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1).
%F A229858 a(n) = n+4 for n>4.
%F A229858 a(n) = 2*a(n-1)-a(n-2) for n>6.
%F A229858 G.f.: -x*(x^5-x^4+x^2+x-3) / (x-1)^2.
%e A229858 12 appears in the sequence because there exists a 120-degree triangle with sides 12, 20 and 28.
%o A229858 (PARI)
%o A229858 \\ Gives values of A not exceeding amax.
%o A229858 \\ e.g. t120a(20) gives [3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
%o A229858 t120a(amax) = {
%o A229858   v=pt120a(amax);
%o A229858   s=[];
%o A229858   for(i=1, #v,
%o A229858     for(m=1, amax\v[i],
%o A229858       if(v[i]*m<=amax, s=concat(s, v[i]*m))
%o A229858     )
%o A229858   );
%o A229858   vecsort(s,,8)
%o A229858 }
%o A229858 \\ Gives values of A not exceeding amax in primitive triangles.
%o A229858 \\ e.g. pt120a(20) gives [3, 5, 7, 9, 11, 13, 15, 16, 17, 19]
%o A229858 pt120a(amax) = {
%o A229858   s=[];
%o A229858   for(m=1, (amax-1)\2,
%o A229858     for(n=1, m-1,
%o A229858       if((m-n)%3!=0 && gcd(m, n)==1,
%o A229858         a=m*m-n*n;
%o A229858         b=n*(2*m+n);
%o A229858         if(a>b, a=b);
%o A229858         if(a<=amax, s=concat(s, a))
%o A229858       )
%o A229858     )
%o A229858   );
%o A229858   vecsort(s,,8)
%o A229858 }
%Y A229858 Cf. A050931, A229849, A229859.
%K A229858 nonn,easy
%O A229858 1,1
%A A229858 _Colin Barker_, Oct 06 2013