

A038589


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


4



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
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OFFSET

0,2


COMMENTS

The hexagonal lattice is the familiar 2dimensional 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
B. 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)) like for the Gauss circle or Dirichlet divisor problems  Benoit Cloitre, Oct 27 2012


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}] (* JeanFrançois 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.
Sequence in context: A143429 A168301 A072821 * A109539 A109541 A173314
Adjacent sequences: A038586 A038587 A038588 * A038590 A038591 A038592


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane.


STATUS

approved



