A002652 Theta series of Kleinian lattice Z[(1 + sqrt(-7))/ 2] in 1 complex (or 2 real) dimensions.

1, 2, 4, 0, 6, 0, 0, 2, 8, 2, 0, 4, 0, 0, 4, 0, 10, 0, 4, 0, 0, 0, 8, 4, 0, 2, 0, 0, 6, 4, 0, 0, 12, 0, 0, 0, 6, 4, 0, 0, 0, 0, 0, 4, 12, 0, 8, 0, 0, 2, 4, 0, 0, 4, 0, 0, 8, 0, 8, 0, 0, 0, 0, 2, 14, 0, 0, 4, 0, 0, 0, 4, 8, 0, 8, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 0, 8, 0, 16, 0, 0, 0, 12, 0, 0, 0, 0, 0, 4, 4, 6, 0

Offset

0,2

Comments

In other words, theta series of lattice with Gram matrix [2, 1; 1, 4].

The number of integer solutions (x, y) to x^2 + x*y + 2*y^2 = n.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 467, Entry 5(i).

Links

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

J. H. Conway and N. J. A. Sloane, Complex and integral laminated lattices, Trans. Amer. Math. Soc., 280 (1983), 463-490.

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

G.f.: theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7).

G.f.: 1 + 2 * Sum_{k>0} Kronecker(-7, k) * x^k / (1 - x^k). - Michael Somos, Mar 17 2012

Expansion of phi(x) * phi(x^7) + 4 * x^2 * psi(x^2) * psi(x^14) = phi(-x) * phi(-x^7) + 4 * x * psi(x) * psi(x^7) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Mar 17 2012

Expansion of ((eta(q) * eta(q^7))^3 + 4 * (eta(q^2) * eta(q^14))^3) / (eta(q) * eta(q^2) * eta(q^7) * eta(q^14)) in powers of q. - Michael Somos, May 28 2005

Moebius transform is period 7 sequence [ 2, 2, -2, 2, -2, -2, 0, ...]. - Michael Somos, Oct 07 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 5 * v^2 + 4 * w^2 + 2 * u*w - 4 * u*v - 8 * v*w. - Michael Somos, Sep 20 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) = u1^3*u6 + 2*u2^3*u3 + 18*u1*u3*u6^2 + 18*u2*u3^2*u6 + 6*u1*u2^2*u6 + 3*u1^2*u2*u3 - 3*u2*u3^3 - 18*u2*u3*u6^2 - 6*u1*u6^3 - 9*u1*u3^2*u6 - 6*u1*u2^2*u3 - 6*u1^2*u2*u6. - Michael Somos, Jun 03 2005

From Michael Somos, Mar 17 2012: (Start)

G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

a(n) = 2 * A035182(n) unless n = 0. a(7*n + 5) = a(7*n + 6) = a(9*n + 3) = a(9*n + 6) = 0. a(2*n + 1) = 2 * A133827(n). a(9*n) = a(n). (End)

a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(7^e) = 1, b(p^e) = e + 1 if p == 1, 2, 4 (mod 7), b(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7). - Michael Somos, Jun 10 2015

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(7) = 2.3748208... . - Amiram Eldar, Dec 16 2023

Example

G.f. = 1 + 2*x + 4*x^2 + 6*x^4 + 2*x^7 + 8*x^8 + 2*x^9 + 4*x^11 + 4*x^14 + ...
Theta series of lattice with Gram matrix [2, 1; 1, 4] = 1 + 2*q^2 + 4*q^4 + 6*q^8 + 2*q^14 + 8*q^16 + 2*q^18 + 4*q^22 + 4*q^28 + 10*q^32 + 4*q^36 + 8*q^44 + 4*q^46 + 2*q^50 + 6*q^56 + 4*q^58 + 12*q^64 + 6*q^72 + ...

Mathematica

f[d_] := KroneckerSymbol[-7, d]; a[n_] := 2*Total[f /@ Divisors[n]]; a[0]=1; Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Nov 08 2011, after Michael Somos *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 Sum[ KroneckerSymbol[ -7, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -7, #] &]]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Length @ FindInstance[ n == x^2 + x y + 2 y^2, {x, y}, Integers, 10^9]]; (* Michael Somos, Jun 10 2015 *)

Prog

(PARI) {a(n) = my(t2, t3); if( n<1, n==0, t2 = 2 * sum( n=1, (sqrtint( max(0, 4*n - 7)) + 1)\2, x^(n*n - n)); t3 = 1 + 2 * sum( n=1, sqrtint(n), x^(n*n)); polcoeff( t3 * subst(t3, x, x^7) + x^2 * t2 * subst(t2, x, x^7), n))};
(PARI) {a(n) = my(t); if( n<1, n==0, 2 * issquare(n) + 2 * sum( y=1, sqrtint(n*4\7), 2 * issquare(t = 4*n - 7*y^2) - (t==0)))}; /* Michael Somos, Sep 20 2004 */
(PARI) {a(n) = my(A, A1, A2); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A) * eta(x^7 + A); A2 = eta(x^2 + A) * eta(x^14 + A); polcoeff( (A1^3 + 4 * x * A2^3) / (A1 * A2), n))}; /* Michael Somos, May 28 2005 */
(PARI) {a(n) = if( n<1, n==0, 2 * qfrep( [ 2, 1; 1, 4], n, 1)[n])}; /* Michael Somos, Jun 03 2005 */
(PARI) {a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -7, d)))}; /* Michael Somos, Oct 07 2005 */
(Magma) A := Basis( ModularForms( Gamma1(14), 1), 85); A[1] + 2*A[2] + 4*A[3] + 6*A[5]; /* Michael Somos, Jun 10 2015 */

Crossrefs

Cf. A035182, A133827, A028951.

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), this sequence (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43).

Sequence in context: A255982 A256061 A323099 * A202541 A070676 A291306

Adjacent sequences: A002649 A002650 A002651 * A002653 A002654 A002655

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved