0,2

One of Ramanujan's 54 universal quaternary quadratic forms. - Michael Somos Apr 01 2008

J. H. Conway and N. J. A. Sloane, Sphere Packing, lattices and Groups, Springer-Verlag, p. 108, Eq. (49).

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.28). See also top of p. 94.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314, Theorem 386.

S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.

S. Ramanujan, Collected Papers, Chap. 20, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1917) 11-21)

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

D. A. Alpern, Proofs of Lagrange 4 square theorem

G. E. Andrews, S. B. Ekhad, D. Zeilberger [math/9206203] A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a Sum of Four Squares

G. E. Andrews, S. B. Ekhad, D. Zeilberger, A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a sum of Four Squares

R. T. Bumby, Sums of four squares, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.

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

E. van Fossen Conrad, Jacobi's Four Square Theorem

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

Simon Plouffe, Table of n, a(n) for n=0..105817

Index entries for sequences related to sums of squares

Y. Mimura, Almost Universal Quadratic Forms.

For n>0, a(n)/8 is multiplicative, and a(p^n)/8 = 1 + p + p^2 + ... + p^n for p an odd prime, a(2^n)/8 = 1 + 2 for n>0.

a(n)=8*A000203(n/A006519(n))*(2+(-1)^n) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 16 2002

G.f.: theta_3(q)^4 = Product( (1-q^(2n))*(1+q^(2n-1))^2, n=1..inf )^4 = eta(-q)^8/eta(q^2)^4; eta = Dedekind's function.

G.f.: 1+8 Sum_{k>0} x^k/(1+(-x)^k)^2 = 1+8 Sum_{k>0} k*x^k/(1+(-x)^k).

G.f. = s(2)^20/(s(1)*s(4))^8, where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]

Fine gives another explicit formula for a(n) in terms of the divisors of n.

8*A046897(n), n>0. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 02 2003

G.f. A(x) satisfies 0=f(A(x), A(x^3), A(x^9)) where f(u, v, w) = v^4 -30*u*v^2*w +12*u*v*w*(u +9*w) -u*w*(u^2 +9*w*u +81*w^2).

G.f. is Fourier series of level 4 weight 2 modular form. f(-1 / (4 t)) = 4 (t/i)^2 f(t) where q = exp(2 pi i t). - Michael Somos, Jan 25 2008

Euler transform of period 4 sequence [ 8, -12, 8, -4, ...].

1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + ...

(add(q^(m^2), m=-10..10))^4;

a[n_] := SumOfSquaresR[4, n]

(PARI) {a(n) = local(A); if( n<0 , 0 , A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / eta(x + A)^2 / eta(x^4 + A)^2)^4, n))}

(PARI) {a(n) = if( n<1, n==0, 8 * sumdiv( n, d, if( d%4, d)))}

A096727(n)=(-1)^n*a(n). A046897(n)=a(n)/8 if n>0. A004011(n)=a(2n). A005879(n)=a(2n+1).

Adjacent sequences: A000115 A000116 A000117 this_sequence A000119 A000120 A000121

Sequence in context: A048109 A068781 A038524 this_sequence A096727 A028660 A028644

nonn,easy,nice

njas