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A033715
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Number of integer solutions (x,y) to the equation x^2+2y^2=n.
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6
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1, 2, 2, 4, 2, 0, 4, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 6, 4, 0, 0, 4, 0, 4, 2, 0, 8, 0, 0, 0, 0, 2, 8, 4, 0, 6, 0, 4, 0, 0, 4, 0, 4, 4, 0, 0, 0, 4, 2, 2, 8, 0, 0, 8, 0, 0, 8, 0, 4, 0, 0, 0, 0, 2, 0, 8, 4, 4, 0, 0, 0, 6, 4, 0, 4, 4, 0, 0, 0, 0, 10, 4, 4, 0, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, 2, 12, 2, 0, 8, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Theta series of lattice C2 with Gram matrix [1,0; 0,2]. a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011
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REFERENCES
| G. E. Andrews, R. Lewis and Z.-G. Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).
B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(iii).
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 19.
J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91.
M. D. Hirschhorn, The number of representations of a number by various forms, Discr. Math., 298 (2005), 205-211.
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LINKS
| John Cannon, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
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FORMULA
| Fine gives an explicit formula for a(n) in terms of the divisors of n.
Euler transform of period 8 sequence [ 2, -1, 2, -4, 2, -1, 2, -2, ...].
Expansion of (eta(q^2) * eta(q^4))^3 / (eta(q) * eta(q^8))^2 in powers of q.
Coefficients in expansion of Sum_{ i, j = -inf .. inf } q^(i^2+2*j^2).
G.f. = s(2)^3*s(4)^3/(s(1)^2*s(8)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
G.f.: 1 + 2 * Sum_{k>0} kronecker(-8, n) * x^k / (1 - x^k) = 1 + 2 * Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(4*k)).
G.f.: theta_3(q) * theta_3(q^2) = Product_{k>0} (1 + x^(2*k)) * ((1 + x^k) * (1 - x^(2*k)) / (1 + x^(4*k)))^2.
Moebius transform is period 8 sequence [ 2, 0, 2, 0, -2, 0, -2, 0, ...]. - Michael Somos, Oct 23 2006
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*u3) * (u1 - u2 - u3 + u6) - (u2 - 3*u6) * (u1 - 2*u2 - u3 + 2*u6). - Michael Somos, Oct 23 2006
a(n) = 2 * A002325(n) unless n=0.
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EXAMPLE
| 1 + 2*q + 2*q^2 + 4*q^3 + 2*q^4 + 4*q^6 + 2*q^8 + 6*q^9 + 4*q^11 + 4*q^12 + ...
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MAPLE
| d:=proc(r, m, n) local i, t1; t1:=0; for i from 1 to n do if n mod i = 0 and i-r mod m = 0 then t1:=t1+1; fi; od: t1; end; [seq(2*(d(1, 8, n)+d(3, 8, n)-d(5, 8, n)-d(7, 8, n)), n=1..120)];
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PROG
| (PARI) a(n) = if(n<=0, n==0, 2*(issquare(n)-issquare(2*n) + 2*sum(i=1, sqrtint(n\2), issquare(n-2*i^2))))
(PARI) {a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -8, d)))} /* Michael Somos, Aug 23 2005 */
(PARI) {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 2], n)[n])} /* Michael Somos, Aug 23 2005 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^3 * eta(x^4 + A)^3 * eta(x^8 + A)^-2, n))}
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CROSSREFS
| Cf. A002325, A002479.
Sequence in context: * A082564 A133692 A139093 A080918 A033758 A033750
Adjacent sequences: A033712 A033713 A033714 * A033716 A033717 A033718
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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