The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A028609 Expansion of (theta_3(z)*theta_3(11z) + theta_2(z)*theta_2(11z)). 15
 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 (list; graph; refs; listen; history; text; internal format)
 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 H. McKean and V. 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 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 Moebius transform is period 11 sequence [ 2, -2, 2, 2, 2, -2, -2, -2, 2, -2, 0, ...]. - Michael Somos, Jan 29 2007 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). - Michael Somos, Jan 29 2007 G.f.: 1 + 2 * Sum_{k>0} Kronecker( -11, k) * x^k / (1 - x^k). - Michael Somos, Jan 29 2007 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. 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. 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 AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 28 00:05 EST 2022. Contains 358406 sequences. (Running on oeis4.)