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

%I A038589 #45 Oct 03 2024 14:06:26
%S A038589 1,7,7,13,19,19,19,31,31,37,37,37,43,55,55,55,61,61,61,73,73,85,85,85,
%T A038589 85,91,91,97,109,109,109,121,121,121,121,121,127,139,139,151,151,151,
%U A038589 151,163,163,163,163,163,169,187,187,187,199,199,199
%N A038589 Sizes of successive clusters in hexagonal lattice A_2 centered at lattice point.
%C A038589 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%H A038589 Vincenzo Librandi, <a href="/A038589/b038589.txt">Table of n, a(n) for n = 0..300</a>
%H A038589 Benoit Cloitre, <a href="https://citeseerx.ist.psu.edu/pdf/b776840ca0750b45bc335d55318f46b1f408820c">On the circle and divisor problems</a>.
%H A038589 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%F A038589 Partial sums of A004016.
%F A038589 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
%F A038589 Equals 1 + A014201(n). - Neven Juric, May 10 2010
%F A038589 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
%F A038589 a(n) = A014201(n) + 1. - _Hugo Pfoertner_, Nov 09 2023
%e A038589 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 + ...
%t A038589 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 *)
%o A038589 (PARI) a(n)=1+6*sum(k=0,n\3,(n\(3*k+1))-(n\(3*k+2)))
%Y A038589 Cf. A004016, A014201, A038589, A038590.
%Y A038589 Cf. A035019.
%Y A038589 Cf. A057655 (for square lattice).
%K A038589 nonn,easy,nice
%O A038589 0,2
%A A038589 _N. J. A. Sloane_