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A014201 Number of solutions to x^2 + x*y + y^2 <= n excluding (0,0).

%I A014201 #32 Sep 03 2025 00:49:58
%S A014201 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 A014201 84,90,90,96,108,108,108,120,120,120,120,120,126,138,138,150,150,150,
%U A014201 150,162,162,162,162,162,168,186,186,186,198,198,198,198,198,210
%N A014201 Number of solutions to x^2 + x*y + y^2 <= n excluding (0,0).
%H A014201 Vincenzo Librandi, <a href="/A014201/b014201.txt">Table of n, a(n) for n = 0..300</a>
%H A014201 Benoit Cloitre, <a href="https://citeseerx.ist.psu.edu/pdf/b776840ca0750b45bc335d55318f46b1f408820c">On the circle and divisor problems</a>.
%F A014201 Equals A038589(n) - 1. - Neven Juric, May 10 2010
%F A014201 From _Benoit Cloitre_, Oct 27 2012: (Start)
%F A014201 a(n) = 6*Sum_{k=0..n/3} (floor(n/(3*k+1)) - floor(n/(3*k+2))).
%F A014201 a(n) is asymptotic to 2*(Pi/sqrt(3))*n.
%F A014201 Conjecture: a(n) = 2*(Pi/sqrt(3))*n + O(n^(1/4 + epsilon)), similar to the Gauss circle or Dirichlet divisor problems. (End)
%t A014201 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 A014201 (PARI) a(n)=6*sum(k=0, n\3, (n\(3*k+1))-(n\(3*k+2))) \\ _Benoit Cloitre_, Oct 27 2012
%Y A014201 Cf. A014202, A014198.
%K A014201 nonn,easy,changed
%O A014201 0,2
%A A014201 _N. J. A. Sloane_