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A028609
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Expansion of (theta_3(z)*theta_3(11z)+theta_2(z)*theta_2(11z)).
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3
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| Theta series of lattice with Gram matrix [2, 1; 1, 6].
Number of integer solutions (x,y) to x^2+xy+3y^2=n. - Michael Somos, Sep 20 2004
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REFERENCES
| H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032)
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LINKS
| John Cannon, Table of n, a(n) for n = 0..5000
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FORMULA
| 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,n)*x^n/(1-x^n) . - Michael Somos Jan 29 2007
G.f. is Fourier series of a weight 1 level 11 modular form. f(-1/ (11 t)) = sqrt(11) (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
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EXAMPLE
| 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 + ...
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PROG
| (PARI) a(n)=local(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<1, n==0, qfrep([2, 1; 1, 6], n, 1)[n]*2) /* Michael Somos Jun 05 2005 */
(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 */
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CROSSREFS
| a(n)=2*A035179(n) if n>0.
Sequence in context: A066910 A094405 A155984 * A107490 A094572 A079534
Adjacent sequences: A028606 A028607 A028608 * A028610 A028611 A028612
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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