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
Examples
G.f. = 1 + 20*x + 180*x^2 + 960*x^3 + 3380*x^4 + 8424*x^5 + 16320*x^6 + ...
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
Programs
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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
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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 *) -
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
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) = (20/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017
Extensions
Extended by Ray Chandler, Nov 28 2006