OFFSET
0,2
COMMENTS
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Benoit Cloitre, On the circle and divisor problems.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
FORMULA
Partial sums of A004016.
Expansion of a(x) / (1 - x) in powers of x where a() is a cubic AGM theta function (cf. A004016). - Michael Somos, Aug 21 2012
Equals 1 + A014201(n). - Neven Juric, May 10 2010
a(n) = 1 + 6*Sum_{k=1..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)) as for the Gauss circle or Dirichlet divisor problems. - Benoit Cloitre, Oct 27 2012
a(n) = A014201(n) + 1. - Hugo Pfoertner, Nov 09 2023
EXAMPLE
1 + 7*x + 7*x^2 + 13*x^3 + 19*x^4 + 19*x^5 + 19*x^6 + 31*x^7 + 31*x^8 + 37*x^9 + ...
MATHEMATICA
a[n_] := 1 + Sum[ Length[ {ToRules[ Reduce[ x^2 + x*y + y^2 == k, {x, y}, Integers] ]}], {k, 1, n}]; Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Feb 23 2012, after Neven Juric *)
PROG
(PARI) a(n)=1+6*sum(k=0, n\3, (n\(3*k+1))-(n\(3*k+2)))
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved