%I A004018 M3218
%S A004018 1,4,4,0,4,8,0,0,4,4,8,0,0,8,0,0,4,8,4,0,8,0,0,0,0,12,8,0,0,8,0,0,4,0,
%T A004018 8,0,4,8,0,0,8,8,0,0,0,8,0,0,0,4,12,0,8,8,0,0,0,0,8,0,0,8,0,0,4,16,0,0,
%U A004018 8,0,0,0,4,8,8,0,0,0,0,0,8,4,8,0,0,16,0,0,0,8,8,0,0,0,0,0,0,8,4,0,12,8
%N A004018 Theta series of square lattice (or number of ways of writing n as a sum 
               of 2 squares).
%C A004018 Number of points in square lattice on the circle of radius sqrt(n).
%C A004018 Often denoted by r(n) or r_2(n).
%C A004018 Let a(n)=A004018(n), b(n)=A004403(n); then Sum(k=1..n)[ a(k)*b(n-k) ] 
               = 0 - John W. Layman (layman(AT)math.vt.edu)
%C A004018 Theta series of D_2 lattice.
%D A004018 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 A004018 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16 (7), r(n).
%D A004018 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 106.
%D A004018 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 78, Eq. (32.23).
%D A004018 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, 
               NY, 1985.
%D A004018 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               3rd ed., Oxford Univ. Press, 1954, p. 240, r(n).
%D A004018 M. D. Hirschhorn, The number of representations of a number by various 
               forms, Discr. Math., 298 (2005), 205-211.
%D A004018 C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. 
               Assoc. Amer., 2000, p. 51.
%D A004018 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A004018 T. D. Noe, <a href="b004018.txt">Table of n, a(n) for n=0..10000</a>
%H A004018 H. H. Chan and C. Krattenthaler, <a href="http://arXiv.org/abs/math.NT/
               0407061">Recent progress in the study of representations of integers 
               as sums of squares</a>
%H A004018 S. Cooper and M. Hirschhorn, <a href="http://www.integers-ejcnt.org/">
               A combinatorial proof of a result from number theory</a>, Integers 
               4 (2004), #A09.
%H A004018 Michael Gilleland, <a href="selfsimilar.html">Some Self-Similar Integer 
               Sequences</a>
%H A004018 M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/
               paper58.pdf">Jacobi's Two-Square Theorem and Related Identities</
               a>
%H A004018 M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/
               paper67.pdf">Arithmetic Consequences of Jacobi's Two-Squares Theorem</
               a>
%H A004018 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               lattices/D2.html">Home page for this lattice</a>
%H A004018 F. Richman, <a href="http://www.math.fau.edu/Richman/gausdisk.htm">Counting 
               Gaussian integers in a disk</a>
%H A004018 G. Villemin, <a href="http://www.multimania.com/villemingerard/Addition/
               NoSoCaPr.htm#Propriete">SOMMES DE PUISSANCES</a>
%H A004018 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MoebiusTransform.html">Link to a section of The World of Mathematics.</
               a>
%H A004018 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SumofSquaresFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A004018 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ThetaSeries.html">Theta Series</a>
%H A004018 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Barnes-WallLattice.html">Barnes-Wall Lattice</a>
%H A004018 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~twosquares.en.html">
               Two squares</a>
%H A004018 <a href="Sindx_Su.html#ssq">Index entries for sequences related to sums 
               of squares</a>
%H A004018 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A004018 Factor n as n = p1^a1 p2^a2 ... q1^b1 q2^b2 ... 2^c, where the p's are 
               primes == 1 mod 4 and the q's are primes == 3 mod 4. Then a(n) = 
               0 if any b is odd, otherwise a(n) = 4 (1 + a1) (1 + a2) ...
%F A004018 Expansion of theta_3(z)^2 = Product_{m >= 1} (1-q^(2m))^2*(1+q^(2m-1))^4.
%F A004018 G.f. = s(2)^10/(s(1)^4*s(4)^4), where s(k) := subs(q=q^k, eta(q)) and 
               eta(q) is Dedekind's function, cf. A010815. [Fine]
%F A004018 Expansion of eta(q^2)^10 / ( eta(q) * eta(q^4) )^4 in powers of q. - 
               Michael Somos, Jul 19 2004
%F A004018 Expansion of ( phi(q)^2 + phi(-q)^2 ) / 2 in powers of q^2 where phi() 
               is a Ramanujan theta function.
%F A004018 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u 
               - v)^2 - (v - w) * 4 * w. - Michael Somos, Jul 19 2004
%F A004018 Euler transform of period 4 sequence [ 4, -6, 4, -2, ...]. - Michael 
               Somos, Jul 19 2004
%F A004018 Moebius transform is period 4 sequence [ 4, 0, -4, 0, ...]. - Michael 
               Somos Sep 17 2007
%F A004018 G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2 (t/
               i) f(t) where q = exp(2 pi i t).
%e A004018 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + 4*q^8 + 4*q^9 + 8*q^10 + 8*q^13 + 4*q^16 
               +
%e A004018 8*q^17 + 4*q^18 + 8*q^20 + 12*q^25 + 8*q^26 + 8*q^29 + 4*q^32 + 8*q^34 
               +
%e A004018 4*q^36 + 8*q^37 + 8*q^40 + 8*q^41 + 8*q^45 + 4*q^49 + 12*q^50 + 8*q^52 
               +
%e A004018 8*q^53 + 8*q^58 + 8*q^61 + 4*q^64 + 16*q^65 + 8*q^68 + 4*q^72 + 8*q^73 
               +
%e A004018 8*q^74 + 8*q^80 + 4*q^81 + 8*q^82 + 16*q^85 + 8*q^89 + 8*q^90 + 8*q^97 
               +
%e A004018 4*q^98 + 12*q^100 + 8*q^101 + 8*q^104 + 8*q^106 + 8*q^109 + 8*q^113 + 
               ... (from John Cannon, Dec 30 2006)
%p A004018 (sum(x^(m^2),m=-10..10))^2;
%t A004018 a[n_] := SumOfSquaresR[2, n]
%t A004018 a[n_] := SquaresR[2, n]
%o A004018 (PARI) a(n)=if(n<0,0,polcoeff(1+4*sum(k=1,n,x^k/(1+x^(2*k)),x*O(x^n)),
               n))
%o A004018 (PARI) a(n)=if(n<1,n==0,4*sumdiv(n,d,(d%4==1)-(d%4==3))) /* Michael Somos, 
               Jul 19 2004 */
%o A004018 (PARI) a(n)=if(n<1,n==0,2*qfrep([1,0;0,1],n)[n]) /* Michael Somos May 
               13 2005 */
%Y A004018 Cf. A001481, A004020, A005883, A057655 (partial sums), A057961, A057962. 
               Except for first term, A004018(n)=4*A002654(n). Partial sums - 1 
               give A014198.
%Y A004018 Cf. A104271, A105673.
%Y A004018 a(n)=A004531(4n). a(n)=2*A105673(n), if n>0.
%Y A004018 Sequence in context: A155836 A164613 A104794 this_sequence A028658 A028642 
               A159796
%Y A004018 Adjacent sequences: A004015 A004016 A004017 this_sequence A004019 A004020 
               A004021
%K A004018 nonn,easy,nice,core
%O A004018 0,2
%A A004018 N. J. A. Sloane (njas(AT)research.att.com).
%E A004018 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000