A028609 Expansion of (theta_3(z)*theta_3(11z) + theta_2(z)*theta_2(11z)).

1, 2, 0, 4, 2, 4, 0, 0, 0, 6, 0, 2, 4, 0, 0, 8, 2, 0, 0, 0, 4, 0, 0, 4, 0, 6, 0, 8, 0, 0, 0, 4, 0, 4, 0, 0, 6, 4, 0, 0, 0, 0, 0, 0, 2, 12, 0, 4, 4, 2, 0, 0, 0, 4, 0, 4, 0, 0, 0, 4, 8, 0, 0, 0, 2, 0, 0, 4, 0, 8, 0, 4, 0, 0, 0, 12, 0, 0, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 8, 0, 0, 0, 4, 0, 6, 6, 0, 0

Offset

0,2

Comments

Theta series of lattice with Gram matrix [2, 1; 1, 6].

Number of integer solutions (x, y) to x^2 + x*y + 3*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

Henry McKean and Victor Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032).

Links

John Cannon, Table of n, a(n) for n = 0..5000

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of phi(x) * phi(x^11) = 4 * x^3 * psi(x^2) * psi(x^22) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 21 2015

From Michael Somos, Jan 29 2007: (Start)

Moebius transform is period 11 sequence [ 2, -2, 2, 2, 2, -2, -2, -2, 2, -2, 0, ...].

a(n) = 2 * b(n) and b(n) is multiplicative with b(11^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), b(p^e) = e + 1 if p == 1, 3, 4, 5, 9 (mod 11).

G.f.: 1 + 2 * Sum_{k>0} Kronecker( -11, k) * x^k / (1 - x^k). (End)

G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 05 2007

Expansion of (F(x)^2 + 4 * F(x^2)^2 + 8 * F(x^4)^2) / F(x^2) in powers of x or expansion of (F(x)^2 + 2 * F(x^2)^2 + 2 * F(x^4)^2) / F(x^2) in powers of x^4 where F(x) = x^(1/2) * f(-x) * f(-x^11) and f() is a Ramanujan theta function. - Michael Somos, Mar 01 2010

a(n) = 2 * A035179(n) unless n=0. Convolution square is A028610.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(11) = 1.894451... . - Amiram Eldar, Dec 16 2023

Example

G.f. = 1 + 2*x + 4*x^3 + 2*x^4 + 4*x^5 + 6*x^9 + 2*x^11 + 4*x^12 + 8*x^15 + ...
Theta series of lattice with Gram matrix [2, 1; 1, 6] = 1 + 2*q^2 + 4*q^6 + 2*q^8 + 4*q^10 + 6*q^18 + 2*q^22 + 4*q^24 + 8*q^30 + 2*q^32 + 4*q^40 + 4*q^46 + 6*q^50 + 8*q^54 + 4*q^62 + 4*q^66 + 6*q^72 + 4*q^74 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[ n == 0], DivisorSum[ n, KroneckerSymbol[ -11, #] &] 2]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^11] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^11], {q, 0, n}]; (* Michael Somos, Jul 12 2014 *)

Prog

(PARI) {a(n) = my(t); if( n<1, n==0, 2 * issquare(n) + 2 * sum( y=1, sqrtint(n * 4\11), 2 * issquare( t=4*n - 11*y^2) - (t==0)))}; /* Michael Somos, Sep 20 2004 */
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep( [ 2, 1; 1, 6], n, 1)), n))}; /* Michael Somos, Apr 21 2015 */
(PARI) {a(n) = if( n<1, n==0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -11, p) * X))[n] * 2)}; /* Michael Somos, Jun 05 2005 */
(PARI) {a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -11, d)))}; /* Michael Somos, Jan 29 2007 */
(Magma) A := Basis( ModularForms( Gamma1(11), 1), 103); A[1] + 2*A[2] + 4*A[4] + 2*A[5]; /* Michael Somos, Jul 12 2014 */

Crossrefs

Cf. A028610, A035179, A028954, A056874.

Cf. A000122, A000700, A010054, A121373.

Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), this sequence (d=-11), A028641 (d=-19), A138811 (d=-43).

Sequence in context: A066910 A094405 A155984 * A107490 A094572 A323905

Adjacent sequences: A028606 A028607 A028608 * A028610 A028611 A028612

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved