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

0,2

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

Cf. A004016, A014201, A038589, A038590.

Cf. A035019.

Cf. A057655 (for square lattice).

Sequence in context: A168301 A335895 A072821 * A332304 A317790 A109539

Adjacent sequences: A038586 A038587 A038588 * A038590 A038591 A038592

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved