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 A033715 Number of integer solutions (x, y) to the equation x^2 + 2y^2 = n. 37

%I

%S 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,

%T 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,

%U 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

%N Number of integer solutions (x, y) to the equation x^2 + 2y^2 = n.

%C 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

%C Number 17 of the 74 eta-quotients listed in Table I of Martin (1996).

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Denoted by |a_4(n)| in Kassel and Reutenauer 2015. - _Michael Somos_, Jun 16 2015

%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(iii).

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

%D 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.

%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).

%D 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.

%D J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 346.

%H John Cannon, <a href="/A033715/b033715.txt">Table of n, a(n) for n = 0..10000</a>

%H G. E. Andrews, R. Lewis and Z.-G. Liu, <a href="http://dx.doi.org/10.1112/blms/33.1.25">An identity relating a theta series to a sum of Lambert series</a>, Bull. London Math. Soc., 33 (2001), 25-31.

%H Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>

%H M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.045">The number of representations of a number by various forms</a>, Discrete Mathematics 298 (2005), 205-211.

%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1505.07229v3">The zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]

%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1610.07793">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1610.07793 [math.NT], 2016.

%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)

%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>

%H M. Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Fine gives an explicit formula for a(n) in terms of the divisors of n.

%F Euler transform of period 8 sequence [ 2, -1, 2, -4, 2, -1, 2, -2, ...].

%F Expansion of (eta(q^2) * eta(q^4))^3 / (eta(q) * eta(q^8))^2 in powers of q.

%F Coefficients in expansion of Sum_{i,j=-inf..inf} q^(i^2 + 2*j^2).

%F 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]

%F G.f.: 1 + 2 * Sum_{k>0} Kronecker(-2, n) * x^k / (1 - x^k) = 1 + 2 * Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(4*k)).

%F 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.

%F Moebius transform is period 8 sequence [ 2, 0, 2, 0, -2, 0, -2, 0, ...]. - _Michael Somos_, Oct 23 2006

%F 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

%F a(n) = 2 * A002325(n) unless n = 0.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - _Michael Somos_, Sep 09 2012

%F Expansion of phi(q) * phi(q^2) in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Aug 29 2014

%F a(2*n) = a(n). a(2*n + 1) = 2 * A113411(n). - _Michael Somos_, Aug 29 2014

%F a(n) = A028572(4*n) = A133692(2*n) = A139093(8*n) = A226225(8*n) = A226240(4*n) = A242609(4*n) = A245572(4*n) / 3 = (-1)^floor((n + 1)/2) * A082564(n). - _Michael Somos_, May 17 2015

%F a(8*n + 5) = a(8*n + 7) = 0. a(8*n + 1) = 2 * A112603(n). a(8*n + 3) = 4 * A033761(n). - _Michael Somos_, May 17 2015

%F a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with a(2^e) = 1, a(p^e) = e + 1 if p == 1, 3 (mod 8), a(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8). - _Jianing Song_, Sep 04 2018

%e G.f. = 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 + ...

%p 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)];

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* _Michael Somos_, Sep 09 2012 *)

%t a[ n_] := If[ n < 1, Boole[ n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* _Michael Somos_, Aug 29 2014 *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^4])^3 / (QPochhammer[ q] QPochhammer[ q^8])^2, {q, 0, n}]; (* _Michael Somos_, Aug 29 2014 *)

%o (PARI) {a(n) = if( n<1, n==0, 2 * (issquare(n) - issquare(2*n) + 2 * sum( i=1, sqrtint(n\2), issquare(n - 2*i^2))))};

%o (PARI) {a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -2, d)))}; /* _Michael Somos_, Aug 23 2005 */

%o (PARI) {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 2], n)[n])}; /* _Michael Somos_, Aug 23 2005 */

%o (PARI) {a(n) = my(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))};

%o (Sage) Q = DiagonalQuadraticForm(ZZ, [1,2]); Q.representation_number_list(104); # _Peter Luschny_, Jun 20 2014

%o (MAGMA) A := Basis( ModularForms( Gamma1(8), 1), 105); A + 2*A + 2*A; /* _Michael Somos_, Aug 29 2014 */

%Y Cf. A002325, A002479, A028572, A033761, A082564, A112603, A113411, A133692, A139093, A226225, A226240, A242609, A245572.

%Y 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), this sequence (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43).

%K nonn

%O 0,2

%A _N. J. A. Sloane_

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Last modified March 29 18:37 EDT 2020. Contains 333117 sequences. (Running on oeis4.)