OFFSET
0,2
COMMENTS
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
a(n) is the number of integer solutions to x^2 + x*y + y^2 = n (or equivalently x^2 - x*y + y^2 = n). - Michael Somos, Sep 20 2004
a(n) is the number of integer solutions to x^2 + y^2 + z^2 = 2*n where x + y + z = 0. - Michael Somos, Mar 12 2012
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) = 6*A002324(n) if n>0, and A002324 is multiplicative, thus a(1)*a(m*n) = a(n)*a(m) if n>0, m>0 are relatively prime. - Michael Somos, Mar 17 2019
The first occurrence of a(n)= 6, 12, 18, 24, ... (multiples of 6) is at n= 1, 7, 49, 91, 2401, 637, 117649, ... (see A002324). - R. J. Mathar, Sep 21 2024
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 171, Entry 28.
Harvey 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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
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).
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.
G. L. Hall, Comment on the paper "Theta series and magic numbers for diamond and certain ionic crystal structures" [J. Math. Phys. 28, 1653 (1987)]. Journal of Mathematical Physics; Sep. 1988, Vol. 29 Issue 9, pp. 2090-2092. - From N. J. A. Sloane, Dec 18 2012
M. D. Hirschhorn, Three classical results on representations of a number, Séminaire Lotharingien de Combinatoire, B42f (1999), 8 pp.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Helmut Ruhland, A family of lattices with an unbounded number of unit vectors, arXiv:2410.16172 [math.MG], 2024. See p. 2.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
N. J. A. Sloane, Tables of Sphere Packings and Spherical Codes, IEEE Trans. Information Theory, vol. IT-27, 1981 pp. 327-338.
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of a(q) in powers of q where a(q) is the first cubic AGM theta function.
Expansion of theta_3(q) * theta_3(q^3) + theta_2(q) * theta_2(q^3) in powers of q.
Expansion of phi(x) * phi(x^3) + 4 * x * psi(x^2) * psi(x^6) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of (1 / Pi) integral_{0 .. Pi/2} theta_3(z, q)^3 + theta_4(z, q)^3 dz in powers of q^2. - Michael Somos, Jan 01 2012
Expansion of coefficient of x^0 in f(x * q, q / x)^3 in powers of q^2 where f(,) is Ramanujan's general theta function. - Michael Somos, Jan 01 2012
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - 3*v^2 - 2*u*w + 4*w^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. 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
G.f. A(x) satisfies A(x) + A(-x) = 2 * A(x^4), from Ramanujan.
G.f.: 1 + 6 * Sum_{k>0} x^k / (1 + x^k + x^(2*k)). - Michael Somos, Oct 06 2003
G.f.: Sum_( q^(n^2+n*m+m^2) ) where the sum (for n and m) extends over the integers. - Joerg Arndt, Jul 20 2011
G.f.: theta_3(q) * theta_3(q^3) + theta_2(q) * theta_2(q^3) = (eta(q^(1/3))^3 + 3 * eta(q^3)^3) / eta(q).
G.f.: 1 + 6*Sum_{n>=1} x^(3*n-2)/(1-x^(3*n-2)) - x^(3*n-1)/(1-x^(3*n-1)). - Paul D. Hanna, Jul 03 2011
a(3*n + 2) = 0, a(3*n) = a(n), a(3*n + 1) = 6 * A033687(n). - Michael Somos, Jul 16 2005
a(2*n + 1) = 6 * A033762(n), a(4*n + 2) = 0, a(4*n) = a(n), a(4*n + 1) = 6 * A112604(n), a(4*n + 3) = 6 * A112595(n). - Michael Somos, May 17 2013
Euler transform of A192733. - Michael Somos, Mar 12 2012
a(n) = (-1)^n * A180318(n). - Michael Somos, Sep 14 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(3) = 3.627598... (A186706). - Amiram Eldar, Oct 15 2022
EXAMPLE
G.f. = 1 + 6*x + 6*x^3 + 6*x^4 + 12*x^7 + 6*x^9 + 6*x^12 + 12*x^13 + 6*x^16 + ...
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 + ...
MAPLE
MATHEMATICA
a[ n_] := If[ n < 1, Boole[ n == 0 ], 6 DivisorSum[ n, KroneckerSymbol[ #, 3] &]]; (* Michael Somos, Nov 08 2011 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / QPochhammer[ q^3], {q, 0, n}]; (* Michael Somos, Nov 13 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3], {q, 0, n}]; (* Michael Somos, Nov 13 2014 *)
a[ n_] := Length @ FindInstance[ x^2 + x y + y^2 == n, {x, y}, Integers, 10^9]; (* Michael Somos, Sep 14 2015 *)
terms = 81; f[q_] = LatticeData["A2", "ThetaSeriesFunction"][-I Log[q]/Pi]; s = Series[f[q], {q, 0, 2 terms}]; CoefficientList[s, q^2][[1 ;; terms]] (* Jean-François Alcover, Jul 04 2017 *)
PROG
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 6 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, p%3==1, e+1, 1-e%2)))}; /* Michael Somos, May 20 2005 */ /* Editor's note: this is the most efficient program */
(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))}; /* Michael Somos, Oct 06 2003 */
(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)))}; /* Michael Somos, May 20 2005 */
(PARI) {a(n) = my(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 */
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 6 * sum( k=1, n, x^(3*k - 2) / (1 - x^(3*k - 2)) - x^(3*k - 1) / (1 - x^(3*k - 1)), x * O(x^n)), n))} /* Paul D. Hanna, Jul 03 2011 */
(Sage) ModularForms( Gamma1(3), 1, prec=81).0 ; # Michael Somos, Jun 04 2013
(Magma) Basis( ModularForms( Gamma1(3), 1), 81) [1]; /* Michael Somos, May 27 2014 */
(Magma) L := Lattice("A", 2); A<q> := ThetaSeries(L, 161); A; /* Michael Somos, Nov 13 2014 */
(Python)
from math import prod
from sympy import factorint
def A004016(n): return 6*prod(e+1 if p%3==1 else int(not e&1) for p, e in factorint(n).items() if p != 3) if n else 1 # Chai Wah Wu, Nov 17 2022
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved