0,2

Number of points in square lattice on the circle of radius sqrt(n).

Often denoted by r(n) or r_2(n).

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)

Theta series of D_2 lattice.

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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16 (7), r(n).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.23).

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 240, r(n).

M. D. Hirschhorn, The number of representations of a number by various forms, Discr. Math., 298 (2005), 205-211.

C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 51.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..10000

H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares

S. Cooper and M. Hirschhorn, A combinatorial proof of a result from number theory, Integers 4 (2004), #A09.

Michael Gilleland, Some Self-Similar Integer Sequences

M. D. Hirschhorn, Jacobi's Two-Square Theorem and Related Identities

M. D. Hirschhorn, Arithmetic Consequences of Jacobi's Two-Squares Theorem

G. Nebe and N. J. A. Sloane, Home page for this lattice

F. Richman, Counting Gaussian integers in a disk

G. Villemin, SOMMES DE PUISSANCES

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Theta Series

Eric Weisstein's World of Mathematics, Barnes-Wall Lattice

G. Xiao, Two squares

Index entries for sequences related to sums of squares

Index entries for "core" sequences

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) ...

Expansion of theta_3(z)^2 = Product_{m >= 1} (1-q^(2m))^2*(1+q^(2m-1))^4.

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]

Expansion of eta(q^2)^10 / ( eta(q) * eta(q^4) )^4 in powers of q. - Michael Somos, Jul 19 2004

Expansion of ( phi(q)^2 + phi(-q)^2 ) / 2 in powers of q^2 where phi() is a Ramanujan theta function.

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

Euler transform of period 4 sequence [ 4, -6, 4, -2, ...]. - Michael Somos, Jul 19 2004

Moebius transform is period 4 sequence [ 4, 0, -4, 0, ...]. - Michael Somos Sep 17 2007

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).

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 +

8*q^17 + 4*q^18 + 8*q^20 + 12*q^25 + 8*q^26 + 8*q^29 + 4*q^32 + 8*q^34 +

4*q^36 + 8*q^37 + 8*q^40 + 8*q^41 + 8*q^45 + 4*q^49 + 12*q^50 + 8*q^52 +

8*q^53 + 8*q^58 + 8*q^61 + 4*q^64 + 16*q^65 + 8*q^68 + 4*q^72 + 8*q^73 +

8*q^74 + 8*q^80 + 4*q^81 + 8*q^82 + 16*q^85 + 8*q^89 + 8*q^90 + 8*q^97 +

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)

(sum(x^(m^2), m=-10..10))^2;

a[n_] := SumOfSquaresR[2, n]

a[n_] := SquaresR[2, n]

(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))

(PARI) a(n)=if(n<1, n==0, 4*sumdiv(n, d, (d%4==1)-(d%4==3))) /* Michael Somos, Jul 19 2004 */

(PARI) a(n)=if(n<1, n==0, 2*qfrep([1, 0; 0, 1], n)[n]) /* Michael Somos May 13 2005 */

Cf. A001481, A004020, A005883, A057655 (partial sums), A057961, A057962. Except for first term, A004018(n)=4*A002654(n). Partial sums - 1 give A014198.

Cf. A104271, A105673.

a(n)=A004531(4n). a(n)=2*A105673(n), if n>0.

Adjacent sequences: A004015 A004016 A004017 this_sequence A004019 A004020 A004021

Sequence in context: A155836 A164613 A104794 this_sequence A028658 A028642 A159796

nonn,easy,nice,core

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000