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

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.

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

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

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

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).

Adjacent sequences: A004013 A004014 A004015 this_sequence A004017 A004018 A004019

Sequence in context: A136526 A097715 A092605 this_sequence A093577 A065442 A055955

nonn,nice,easy

njas