A138806 Expansion of (theta_3(q) * theta_3(q^27) + theta_2(q) * theta_2(q^27) - 1) / 2 in powers of q.

1, 0, 0, 1, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 6, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0

Offset

1,7

Comments

Half the number of integer solutions to x^2 + x*y + 7*y^2 = n. - Jianing Song, Nov 20 2019

Links

Table of n, a(n) for n=1..105.

Formula

a(n) is multiplicative and a(3^e) = 3 if e>1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).

a(3*n + 2) = a(4*n + 2) = 0.

G.f.: (Sum_{i,j} x^(i*i + i*j + 7*j*j) - 1) / 2.

A138805(n) = 2 * a(n) unless n=0. A033687(n) = a(3*n + 1). A097195(n) = a(6*n + 1). A123884(n) = a(12*n + 1). 2 * A121361(n) = a(12*n + 7).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 16 2023

Example

q + q^4 + 2*q^7 + 3*q^9 + 2*q^13 + q^16 + 2*q^19 + q^25 + 3*q^27 + ...

Mathematica

f[p_, e_] := If[Mod[p, 6] == 1, e + 1, (1 + (-1)^e)/2]; f[2, e_] := 1 - Mod[e, 2]; f[3, e_] := 3; f[3, 1] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 07 2023 *)

Prog

(PARI) {a(n) = if( n<1, 0, if( n%3 == 2, 0, if( n%3==1, sumdiv(n, d, kronecker(-3, d)), if( n%9==0, 3 * sumdiv(n/9, d, kronecker(-3, d))))))}
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-3, d)) - if( n%3==0, sumdiv(n/3, d, [0, 1, -1, -3, 1, -1, 3, 1, -1][d%9+1])))}
(PARI) {a(n) = if( n<1, 0, qfrep([2, 1; 1, 14], n, 1)[n])}

Crossrefs

Cf. A138805 (number of integer solutions to x^2 + x*y + 7*y^2 = n).

Cf. A033687, A073010, A097195, A123884, A121361.

Similar sequences: A096936, A113406, A110399.

Sequence in context: A116864 A255308 A079302 * A181105 A142971 A104117

Adjacent sequences: A138803 A138804 A138805 * A138807 A138808 A138809

Keyword

nonn,easy,mult

Author

Michael Somos, Mar 30 2008

Status

approved