%I #28 Aug 06 2024 05:28:24
%S 0,6,6,12,18,18,18,30,30,36,36,36,42,54,54,54,60,60,60,72,72,84,84,84,
%T 84,90,90,96,108,108,108,120,120,120,120,120,126,138,138,150,150,150,
%U 150,162,162,162,162,162,168
%N Number of solutions to x^2 + x*y + y^2 <= n excluding (0,0).
%H Vincenzo Librandi, <a href="/A014201/b014201.txt">Table of n, a(n) for n = 0..300</a>
%H Benoit Cloitre, <a href="https://citeseerx.ist.psu.edu/pdf/b776840ca0750b45bc335d55318f46b1f408820c">On the circle and divisor problems</a>.
%F Equals A038589(n) - 1. - Neven Juric, May 10 2010
%F From _Benoit Cloitre_, Oct 27 2012: (Start)
%F a(n) = 6*Sum_{k=0..n/3} (floor(n/(3k+1)) - floor(n/(3k+2))).
%F a(n) is asymptotic to 2*(Pi/sqrt(3))*n.
%F Conjecture: a(n) = 2*(Pi/sqrt(3))*n + O(n^(1/4 + epsilon)), similar to the Gauss circle or Dirichlet divisor problems. (End)
%t a[n_] := Sum[ Length[ {ToRules[ Reduce[ x^2 + x*y + y^2 == k, {x, y}, Integers]]}], {k, 1, n}]; Table[ a[n], {n, 0, 48}] (* _Jean-François Alcover_, Feb 23 2012 *)
%o (PARI) a(n)=6*sum(k=0, n\3, (n\(3*k+1))-(n\(3*k+2))) \\ Benoit Cloitre, Oct 27 2012
%Y Cf. A014202, A014198.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_