A000143 Number of ways of writing n as a sum of 8 squares.

1, 16, 112, 448, 1136, 2016, 3136, 5504, 9328, 12112, 14112, 21312, 31808, 35168, 38528, 56448, 74864, 78624, 84784, 109760, 143136, 154112, 149184, 194688, 261184, 252016, 246176, 327040, 390784, 390240, 395136, 476672, 599152, 596736, 550368, 693504, 859952

Offset

0,2

Comments

The relevant identity for the o.g.f. is theta_3(x)^8 = 1 + 16*Sum_{j >= 1} j^3*x^j/(1 - (-1)^j*x^j). See the Hardy-Wright reference, p. 315. - Wolfdieter Lang, Dec 08 2016

References

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61); p. 79 Eq. (32.32).

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, pp. 314 - 315.

Links

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

J. M. Borwein and K.-K. S. Choi, On Dirichlet series for sums of squares, Ramanujan J. 7 (1-3) (2003) 95-127.

H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.

Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

P. J. C. Lamont, The number of Cayley integers of given norm, Proceedings of the Edinburgh Mathematical Society, 25.1 (1982): 101-103. See (5).

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.

M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for sequences related to sums of squares

Formula

Expansion of theta_3(z)^8. Also a(n)=16*(-1)^n*Sum_{0<d|n} (-1)^d*d^3.

Expansion of phi(q)^8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 21 2008

Expansion of (eta(q^2)^5 / (eta(q) * eta(q^4))^2)^8 in powers of q. - Michael Somos, Sep 25 2005

G.f.: s(2)^40/(s(1)*s(4))^16, where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]

Euler transform of period 4 sequence [16, -24, 16, -8, ...]. - Michael Somos, Apr 10 2005

a(n) = 16 * b(n) and b(n) is multiplicative with b(p^e) = (p^(3*e+3) - 1) / (p^3 - 1) -2[p<3]. - Michael Somos, Sep 25 2005

G.f.: 1 + 16 * Sum_{k>0} k^3 * x^k / (1 - (-x)^k). - Michael Somos, Sep 25 2005

A035016(n) = (-1)^n * a(n). 16 * A008457(n) = a(n) unless n=0.

Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = 16*(1 - 2^(1-s) + 4^(2-s))*zeta(s)*zeta(s-3). [Borwein and Choi], R. J. Mathar, Jul 02 2012

a(n) = (16/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Example

1 + 16*q + 112*q^2 + 448*q^3 + 1136*q^4 + 2016*q^5 + 3136*q^6 + 5504*q^7 + ...

Maple

(sum(x^(m^2), m=-10..10))^8;
with(numtheory); rJ := n-> if n=0 then 1 else 16*add((-1)^(n+d)*d^3, d in divisors(n)); fi; [seq(rJ(n), n=0..100)]; # N. J. A. Sloane, Sep 15 2018

Mathematica

Table[SquaresR[8, n], {n, 0, 33}] (* Ray Chandler, Dec 06 2006 *)
SquaresR[8, Range[0, 50]] (* Harvey P. Dale, Aug 26 2011 *)
QP = QPochhammer; s = (QP[q^2]^5/(QP[q]*QP[q^4])^2)^8 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

Prog

(PARI) {a(n) = if( n<1, n==0, 16 * (-1)^n * sumdiv( n, d, (-1)^d * d^3))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^8, n))} /* Michael Somos, Sep 25 2005 */
(Sage)
Q = DiagonalQuadraticForm(ZZ, [1]*8)
Q.representation_number_list(60) # Peter Luschny, Jun 20 2014
(Julia) # JacobiTheta3 is defined in A000122.
A000143List(len) = JacobiTheta3(len, 8)
A000143List(37) |> println # Peter Luschny, Mar 12 2018

Crossrefs

8th column of A286815. - Seiichi Manyama, May 27 2017

Row d=8 of A122141.

Cf. A008457, A035016, A010815.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cf. A004018, A000118, A000141 for the expansion of the powers of 2, 4, 6 of theta_3(x).

Sequence in context: A279425 A144449 A035016 * A258546 A205964 A222113

Adjacent sequences: A000140 A000141 A000142 * A000144 A000145 A000146

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved