Search: id:A033715
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%I A033715
%S A033715 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,
%T A033715 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,
%U A033715 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 A033715 Number of integer solutions (x,y) to the equation x^2+2y^2=n.
%C A033715 Theta series of lattice C2 with Gram matrix [1,0; 0,2].
%C A033715 Euler transform of period 8 sequence [2,-1,2,-4,2,-1,2,-2,...].
%C A033715 Expansion of (eta(q^2)eta(q^4))^3/(eta(q)eta(q^8))^2 in powers of q.
%D A033715 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.
%D A033715 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p 102 eq 9.
%D A033715 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 78, Eq. (32.24).
%D A033715 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 
               114 Entry 8(iii).
%D A033715 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 A033715 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 A033715 M. D. Hirschhorn, The number of representations of a number by various 
               forms, Discr. Math., 298 (2005), 205-211.
%H A033715 John Cannon, <a href="b033715.txt">Table of n, a(n) for n = 0..10000</
               a>
%H A033715 Michael Gilleland, <a href="selfsimilar.html">Some Self-Similar Integer 
               Sequences</a>
%F A033715 Fine gives an explicit formula for a(n) in terms of the divisors of n.
%F A033715 Coefficients in expansion of Sum_{ i, j = -inf .. inf } q^(i^2+2*j^2).
%F A033715 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 A033715 G.f.: 1+2 Sum_{k>0} kronecker(-8, n)x^k/(1-x^k) = 1+2 Sum_{k>0} (x^k+x^(3k))/
               (1+x^(4k)).
%F A033715 G.f.: theta_3(q)theta_3(q^2) = Product_{k>0} (1+x^(2k))((1+x^k)(1-x^(2k))/
               (1+x^(4k)))^2.
%F A033715 Moebius transform is period 8 sequence [ 2, 0, 2, 0, -2, 0, -2, 0, ...]. 
               - Michael Somos Oct 23 2006
%F A033715 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
%p A033715 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)];
%o A033715 (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))))
%o A033715 (PARI) a(n)=if(n<1, n==0, 2*sumdiv(n,d,kronecker(-8,d))) /* Michael Somos 
               Aug 23 2005 */
%o A033715 (PARI) a(n)=if(n<1, n==0, 2*qfrep([1,0;0,2],n)[n]) /* Michael Somos Aug 
               23 2005 */
%o A033715 (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))}
%Y A033715 a(n)=2*A002325(n) for n>0.
%Y A033715 Sequence in context: A114427 A129355 A080963 this_sequence A082564 A133692 
               A139093
%Y A033715 Adjacent sequences: A033712 A033713 A033714 this_sequence A033716 A033717 
               A033718
%K A033715 nonn
%O A033715 0,2
%A A033715 N. J. A. Sloane (njas(AT)research.att.com).

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