%I A000122
%S A000122 1,2,0,0,2,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,
%T A000122 0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,
%U A000122 0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0
%N A000122 Expansion of Jacobi theta function theta_3(x) = Sum_{m = -infinity..infinity} 
               x^(m^2) (number of solutions to k^2 = n).
%C A000122 Theta series of the one-dimensional lattice Z: 1 + 2*q + 2*q^4 + 2*q^9 
               + 2*q^16 + 2*q^25 + 2*q^36 + 2*q^49 + 2*q^64 + 2*q^81 + 2*q^100 + 
               ...
%C A000122 Also, essentially the same as the theta series of the one-dimensional 
               lattices A_1, A*_1, D_1, D*_1.
%C A000122 Number of ways of writing n as a square.
%C A000122 Closely related: theta_4(x) = Sum_{m = -infinity..infinity} (-x)^(m^2).
%C A000122 Euler transform of period 4 sequence [2,-3,2,-1,...].
%C A000122 Expansion of eta(q^2)^5/(eta(q)eta(q^4))^2 in powers of q.
%C A000122 G.f. A(x) satisfies 0=f(A(x),A(x^2),A(x^4)) where f(u,v,w)=u^2-v^2+2w(w-u). 
               - Michael Somos, Jul 20 2004
%D A000122 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 64.
%D A000122 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 104, [5n].
%D A000122 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 102.
%D A000122 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 93, Eq. (34.1); p. 78, Eq. (32.22).
%D A000122 G. H. Hardy and E. M. Wright, Theorem 352, p. 282.
%D A000122 E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge 
               Univ. Press, 4th ed., 1963, p. 464.
%H A000122 T. D. Noe, <a href="b000122.txt">Table of n, a(n) for n=0..10000</a>
%H A000122 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A000122 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A000122 Sum(x^(m^2), m=-infinity..infinity);
%F A000122 a(0) = 1; for n >= 0, a(n) = 0 unless n is a square when a(n) = 2.
%F A000122 G.f.: Product_{k>0} (1-x^(2k)) (1+x^(2k-1))^2.
%F A000122 G.f. = s(2)^5/(s(1)^2*s(4)^2), where s(k) := subs(q=q^k, eta(q)), where 
               eta(q) is Dedekind's function, cf. A010815. [Fine]
%F A000122 The Jacobi triple product identity states that for |x| < 1, z != 0, Product_{n>
               0} {(1-x^(2n))(1+x^(2n-1)z)(1+x^(2n-1)/z)} = Sum_{n= -inf..inf} x^(n^2)z^n. 
               Set z=1 to get theta_3(x).
%p A000122 add(x^(m^2),m=-10..10);
%t A000122 CoefficientList[ Sum[ x^(m^2), {m, -(n=10), n} ], x ]
%o A000122 (PARI) a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(-X)^2/eta(X^2),
               n))
%o A000122 (PARI) a(n)=issquare(n)*2-(n==0)
%Y A000122 Cf. A002448. Partial sums give A001650.
%Y A000122 Sequence in context: A093492 A128771 A139380 this_sequence A002448 A033759 
               A033755
%Y A000122 Adjacent sequences: A000119 A000120 A000121 this_sequence A000123 A000124 
               A000125
%K A000122 nonn,easy,nice
%O A000122 0,2
%A A000122 N. J. A. Sloane (njas(AT)research.att.com).