A014201 Number of solutions to x^2 + x*y + y^2 <= n excluding (0,0).
0, 6, 6, 12, 18, 18, 18, 30, 30, 36, 36, 36, 42, 54, 54, 54, 60, 60, 60, 72, 72, 84, 84, 84, 84, 90, 90, 96, 108, 108, 108, 120, 120, 120, 120, 120, 126, 138, 138, 150, 150, 150, 150, 162, 162, 162, 162, 162, 168, 186, 186, 186, 198, 198, 198, 198, 198, 210
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..300
- Benoit Cloitre, On the circle and divisor problems.
Programs
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Mathematica
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 *) -
PARI
a(n)=6*sum(k=0, n\3, (n\(3*k+1))-(n\(3*k+2))) \\ Benoit Cloitre, Oct 27 2012
Formula
Equals A038589(n) - 1. - Neven Juric, May 10 2010
From Benoit Cloitre, Oct 27 2012: (Start)
a(n) = 6*Sum_{k=0..n/3} (floor(n/(3*k+1)) - floor(n/(3*k+2))).
a(n) is asymptotic to 2*(Pi/sqrt(3))*n.
Conjecture: a(n) = 2*(Pi/sqrt(3))*n + O(n^(1/4 + epsilon)), similar to the Gauss circle or Dirichlet divisor problems. (End)