A000144 Number of ways of writing n as a sum of 10 squares.

1, 20, 180, 960, 3380, 8424, 16320, 28800, 52020, 88660, 129064, 175680, 262080, 386920, 489600, 600960, 840500, 1137960, 1330420, 1563840, 2050344, 2611200, 2986560, 3358080, 4194240, 5318268, 5878440, 6299520, 7862400, 9619560

Offset

0,2

References

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.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135 section 9.3. MR0106147 (21 #4881)

Links

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, 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.

J. Liouville, Nombre des représentations d’un entier quelconque sous la forme d’une somme de dix carrés, Journal de mathématiques pures et appliquées 2e série, tome 11 (1866), p. 1-8.

Index entries for sequences related to sums of squares

Formula

Euler transform of period 4 sequence [ 20, -30, 20, -10, ...]. - Michael Somos, Sep 12 2005

Expansion of eta(q^2)^50 / (eta(q) * eta(q^4))^20 in powers of q. - Michael Somos, Sep 12 2005

a(n) = 4/5 * (A050456(n) + 16*A050468(n) + 8*A030212(n)) if n>0. - Michael Somos, Sep 12 2005

a(n) = (20/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Example

G.f. = 1 + 20*x + 180*x^2 + 960*x^3 + 3380*x^4 + 8424*x^5 + 16320*x^6 + ...

Maple

(sum(x^(m^2), m=-10..10))^10;
# Alternative:
A000144list := proc(len) series(JacobiTheta3(0, x)^10, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000144list(30); # Peter Luschny, Oct 02 2018

Mathematica

Table[SquaresR[10, n], {n, 0, 30}] (* Ray Chandler, Jun 29 2008; updated by T. D. Noe, Jan 23 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^10, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^k)^10 * (1 + x^k)^30 / (1 + x^(2*k))^20, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 24 2017 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^10, n))}; /* Michael Somos, Sep 12 2005 */
(Sage)
Q = DiagonalQuadraticForm(ZZ, [1]*10)
Q.representation_number_list(37) # Peter Luschny, Jun 20 2014

Crossrefs

Row d=10 of A122141 and of A319574, 10th column of A286815.

Sequence in context: A091983 A037250 A361569 * A361609 A219581 A177073

Adjacent sequences: A000141 A000142 A000143 * A000145 A000146 A000147

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Extended by Ray Chandler, Nov 28 2006

Status

approved