|
| |
|
|
A004016
|
|
Theta series of planar hexagonal lattice A_2.
(Formerly M4042)
|
|
32
| |
|
|
1, 6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0, 0, 6, 0, 0, 12, 0, 12, 0, 0, 0, 6, 0, 6, 12, 0, 0, 12, 0, 0, 0, 0, 6, 12, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 6, 18, 0, 0, 12, 0, 0, 0, 0, 12, 0, 0, 0, 12, 0, 12, 6, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 6, 12, 0, 0, 12, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 0, 24, 0, 12
(list; graph; refs; listen; history; internal format)
|
|
|
|
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.
The number of integer solutions (x,y) to x^2 + x*y + y^2 = n. - Michael Somos, Sep 20 2004
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(
q) (A010054), chi(q) (A000700).
|
|
|
REFERENCES
| 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
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| 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
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to A2 = hexagonal = triangular lattice
|
|
|
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 a Ramanujan 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 +3eta(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)). [From Paul D. Hanna, Jul 3 2011]
a(3*n + 2) = 0, a(3*n) = a(n), a(3*n + 1) = 6 * A033687(n). - Michael Somos, Jul 16 2005
a(n) = 6 * A002324(n) if n>0. a(n) = A005928(3*n).
|
|
|
EXAMPLE
| 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 + ...
|
|
|
MATHEMATICA
| a[ n_] := If[ n < 1, Boole[ n == 0 ], 6 DivisorSum[ n, KroneckerSymbol[ #, 3] &]] (* Michael Somos, Nov 08 2011 *)
|
|
|
PROG
| (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 */
(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 3 2011 */
|
|
|
CROSSREFS
| Cf. A002324, A003051, A003215, A005881, A005882, A005928, A008458, A033685, A033687, A038587-A038591 etc.
See also A035019.
Sequence in context: A198499 A092605 * A180318 A093577 A065442 A198752
Adjacent sequences: A004013 A004014 A004015 * A004017 A004018 A004019
|
|
|
KEYWORD
| nonn,nice,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
