A038589 Sizes of successive clusters in hexagonal lattice A_2 centered at lattice point.

1, 7, 7, 13, 19, 19, 19, 31, 31, 37, 37, 37, 43, 55, 55, 55, 61, 61, 61, 73, 73, 85, 85, 85, 85, 91, 91, 97, 109, 109, 109, 121, 121, 121, 121, 121, 127, 139, 139, 151, 151, 151, 151, 163, 163, 163, 163, 163, 169, 187, 187, 187, 199, 199, 199

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Examples

			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 + ...
		

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

Crossrefs

Cf. A004016, A014201, A038589, A038590.

Cf. A035019.

Cf. A057655 (for square lattice).

Programs

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 *)

PARI

a(n)=1+6*sum(k=0,n\3,(n\(3*k+1))-(n\(3*k+2)))

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