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A000144 Number of ways of writing n as a sum of 10 squares. 4
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 (list; graph; refs; listen; history; text; internal format)
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;

MATHEMATICA

Table[SquaresR[10, n], {n, 0, 30}] (* Ray Chandler *)

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

10th column of A286815. - Seiichi Manyama, May 27 2017

Row d=10 of A122141.

Sequence in context: A159538 A091983 A037250 * A219581 A177073 A211153

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

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Last modified November 20 04:05 EST 2017. Contains 294959 sequences.