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

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Benoit Cloitre, On the circle and divisor problems.

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/(3k+1)) - floor(n/(3k+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)

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 *)

Prog

(PARI) a(n)=6*sum(k=0, n\3, (n\(3*k+1))-(n\(3*k+2))) \\ Benoit Cloitre, Oct 27 2012

Crossrefs

Cf. A014202, A014198.

Sequence in context: A315787 A309458 A029682 * A315788 A315789 A315790

Adjacent sequences: A014198 A014199 A014200 * A014202 A014203 A014204

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved