A014201 -id:A014201 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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 + ...

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

Cf. A004016, A014201, A038589, A038590.
Cf. A035019.
Cf. A057655 (for square lattice).

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 *)

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

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

A014202 Number of solutions to x^2 + x*y + y^2 <= n, excluding (0,0), divided by 6.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 3, 5, 5, 6, 6, 6, 7, 9, 9, 9, 10, 10, 10, 12, 12, 14, 14, 14, 14, 15, 15, 16, 18, 18, 18, 20, 20, 20, 20, 20, 21, 23, 23, 25, 25, 25, 25, 27, 27, 27, 27, 27, 28, 31, 31, 31, 33, 33, 33, 33, 33, 35, 35, 35

Offset: 0

N. J. A. Sloane

A014201/6. Cf. A014200.

a(n)=if(n<1,0,sum(k=1,n,kronecker(k,3)*floor(n/k))) \\ Benoit Cloitre, Oct 31 2009

For n > 0, a(n) = Sum_{k=1..n} K(k,3)*floor(n/k) where K(x,y) is the Kronecker symbol. - Benoit Cloitre, Oct 31 2009

A307109 Number of integer solutions to x^2 + x*y + y^2 <= n excluding (0,0), divided by 2.

Original entry on oeis.org

1, 1, 4, 5, 5, 5, 11, 11, 12, 12, 12, 15, 21, 21, 21, 22, 22, 22, 28, 28, 34, 34, 34, 34, 35, 35, 38, 44, 44, 44, 50, 50, 50, 50, 50, 51, 57, 57, 63, 63, 63, 63, 69, 69, 69, 69, 69, 72, 79, 79, 79, 85, 85, 85, 85, 85, 91, 91, 91, 91, 97, 97, 103, 104, 104, 104

Offset: 1

Hugo Pfoertner, Mar 25 2019

nonn

			a(1)=a(2)=(2/2)=1: 2 solutions (-1,1),(1,-1);
a(3)=(8/2)=4: 2 solutions for n<3 and the 6 solutions (1,1), (-1,-1), (1,-2), (2,-1), (-1,2), (-2,1);
a(4)=a(5)=a(6)=10/2=5: 8 solutions for n<6 and the 2 solutions (-2,2), (2,-2).

Cf. A014201, A014202, A198799, A230655, A300419.

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

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A014202 Number of solutions to x^2 + x*y + y^2 <= n, excluding (0,0), divided by 6.

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A307109 Number of integer solutions to x^2 + x*y + y^2 <= n excluding (0,0), divided by 2.

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