OFFSET
1,3
COMMENTS
This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661. - Michael Somos, Jun 07 2015
Half of the number of integer solutions to x^2 + x*y + 3*y^2 = n. - Michael Somos, Jun 05 2005
From Jianing Song, Sep 07 2018: (Start)
Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -11.
Inverse Moebius transform of A011582. (End)
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -11. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022
REFERENCES
Henry McKean and Victor Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
FORMULA
a(n) is multiplicative with a(11^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = e + 1 if p == 1, 3, 4, 5, 9 (mod 11). - Michael Somos, Jan 29 2007
Moebius transform is period 11 sequence [ 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 0, ...]. - Michael Somos, Jan 29 2007
G.f.: Sum_{k>0} Kronecker(-11, k) * x^k / (1 - x^k). - Michael Somos, Jan 29 2007
A028609(n) = 2 * a(n) unless n = 0. - Michael Somos, Jun 24 2011
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(11) = 0.947225... . - Amiram Eldar, Oct 11 2022
EXAMPLE
G.f. = x + 2*x^3 + x^4 + 2*x^5 + 3*x^9 + x^11 + 2*x^12 + 4*x^15 + x^16 + 2*x^20 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -11, #] &]]; (* Michael Somos, Jun 07 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, qfrep([2, 1; 1, 6], n, 1)[n])}; \\ Michael Somos, Jun 05 2005
(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -11, p)*X))) [n])}; \\ Michael Somos, Jun 05 2005
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -11, d)))};
(Magma) A := Basis( ModularForms( Gamma1(11), 1), 88); B<q> := (-1 + A[1] + 2*A[2] + 4*A[4] + 2*A[5]) / 2; B; // Michael Somos, Jun 07 2015
CROSSREFS
Moebius transform gives A011582.
Cf. A056874 (primes not inert in Q(sqrt(-11))), A296920 (primes decomposing), A191060 (primes remaining inert).
Dedekind zeta functions for imaginary quadratic number fields of discriminants D = -3..-47, -67, -163: A002324, A002654, A035182, A002325, this sequence, A035175, A035171, A035170, A035167, A192013, A035159, A035155, A035151, A035180, A035147, A035143, A318982, A318983.
KEYWORD
nonn,easy,mult
AUTHOR
STATUS
approved