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A038589
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Sizes of successive clusters in hexagonal lattice A_2 centered at lattice point.
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4
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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
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0,2
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COMMENTS
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The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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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 + ...
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=1+6*sum(k=0, n\3, (n\(3*k+1))-(n\(3*k+2)))
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CROSSREFS
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Cf. A004016, A014201, A038589, A038590.
Sequence in context: A168301 A335895 A072821 * A332304 A317790 A109539
Adjacent sequences: A038586 A038587 A038588 * A038590 A038591 A038592
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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