0,2

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

The number of integer solutions (x,y) to x^2+xy+y^2=n. - Michael Somos, Sep 20 2004

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc., 128 (1999), 1333-1338; F_3(q).

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 171, Entry 28.

J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 695.

H. Cohn, Advanced Number Theory, Dover Publications, Inc., 1980, p. 89. Ex. 18.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.

M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 236.

N. J. A. Sloane, Tables of Sphere Packings and Spherical Codes, IEEE Trans. Information Theory, vol. IT-27, 1981 pp. 327-338

T. D. Noe, Table of n, a(n) for n=0..1000

M. D. Hirschhorn, Three classical results on representations of a number

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Index entries for sequences related to A2 = hexagonal = triangular lattice

G.f. A(x) satisfies A(x)+A(-x)=2A(x^4), from Ramanujan.

G.f.: theta_3(q)*theta_3(q^3)+theta_2(q)*theta_2(q^3).

G.f.: 1+6*Sum_{k>0} x^k/(1+x^k+x^(2*k)). - Michael Somos, Oct 06, 2003

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2-3v^2-2uw+4w^2 . - Michael Somos Jun 11 2004

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=(u1-u3)(u3-u6)-(u2-u6)^2 . - Michael Somos May 20 2005

G.f.: theta_3(q)*theta_3(q^3)+theta_2(q)*theta_2(q^3) = (eta(q^(1/3))^3 +3eta(q^3)^3)/eta(q).

a(3n+2)=0, a(3n)=a(n), a(3n+1)=6 A033687(n). - Michael Somos, Jul 16 2005

Expansion of a(q) in powers of q where a(q) is the first cubic AGM analog function.

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^(1/2) (t/i) f(t) where q = exp(2 pi i t). - Michael Somos Sep 11 2007

Theta series of A_2 on the standard scale in which the minimal norm is 2:

1 + 6*q^2 + 6*q^6 + 6*q^8 + 12*q^14 + 6*q^18 + 6*q^24 + 12*q^26 + 6*q^32 +

12*q^38 + 12*q^42 + 6*q^50 + 6*q^54 + 12*q^56 + 12*q^62 + 6*q^72 + 12*q^74 +

12*q^78 + 12*q^86 + 6*q^96 + 18*q^98 + 12*q^104 + 12*q^114 + 12*q^122 +

12*q^126 + 6*q^128 + 12*q^134 + 12*q^146 + 6*q^150 + 12*q^152 + 12*q^158 + ...

(PARI) a(n)=if(n<0, 0, polcoeff(1+6*sum(k=1, n, x^k/(1+x^k+x^(2*k)), x*O(x^n)), n))

(PARI) a(n)=if(n<1, n==0, 6*sumdiv(n, d, kronecker(d, 3))) /* Michael Somos Mar 16 2005 */

(PARI) a(n)=if(n<1, n==0, 6*sumdiv(n, d, (d%3==1)-(d%3==2)))

(PARI) {a(n)=local(A, p, e); if(n<1, n==0, A=factor(n); 6*prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3, 1, if(p%3==1, e+1, !(e%2))))))} /* Michael Somos May 20 2005 */

(PARI) {a(n)=local(A); if(n<0, 0, n*=3; A=x*O(x^n); polcoeff( (eta(x+A)^3+3*x*eta(x^9+A)^3)/eta(x^3+A), n))} /* Michael Somos May 20 2005 */

(PARI) a(n)=if(n<1, n==0, qfrep([2, 1; 1, 2], n, 1)[n]*2) /* Michael Somos Jul 16 2005 */

Cf. A003051, A003215, A005881, A005882, A008458, A033685, A038587-A038591 etc.

See also A035019.

a(n)=6 A002324(n) if n>0. a(n)=A005928(3n).

Sequence in context: A136526 A097715 A092605 this_sequence A093577 A065442 A141462

Adjacent sequences: A004013 A004014 A004015 this_sequence A004017 A004018 A004019

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).