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 A000141 Number of ways of writing n as a sum of 6 squares. 17

%I

%S 1,12,60,160,252,312,544,960,1020,876,1560,2400,2080,2040,3264,4160,

%T 4092,3480,4380,7200,6552,4608,8160,10560,8224,7812,10200,13120,12480,

%U 10104,14144,19200,16380,11520,17400,24960,18396,16440,24480,27200

%N Number of ways of writing n as a sum of 6 squares.

%C The relevant identity for the o.g.f. is theta_3(x)^6 = 1 + 16*Sum_{j>=1} j^2*x^j/(1 + x^(2*j)) - 4*Sum_{j >=0} (-1)^j*(2*j+1)^2 *x^(2*j+1)/(1 - x^(2*j+1)), See the Hardy-Wright reference, p. 315, first equation. - _Wolfdieter Lang_, Dec 08 2016

%D Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, 2013, Preprint submitted to Journal of Geometry and Physics.

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

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

%H T. D. Noe, <a href="/A000141/b000141.txt">Table of n, a(n) for n = 0..10000</a>

%H L. Carlitz, <a href="http://dx.doi.org/10.1090/S0002-9939-1957-0084520-2">Note on sums of four and six squares</a>, Proc. Amer. Math. Soc. 8 (1957), 120-124

%H S. H. Chan, <a href="http://www.jstor.org/stable/4145192">An elementary proof of Jacobi's six squares theorem</a>, Amer. Math. Monthly, 111 (2004), 806-811.

%H H. H. Chan and C. Krattenthaler, <a href="http://arXiv.org/abs/math.NT/0407061">Recent progress in the study of representations of integers as sums of squares</a>, arXiv:math/0407061 [math.NT], 2004.

%H Shi-Chao Chen, <a href="http://dx.doi.org/10.1016/j.jnt.2010.01.011">Congruences for rs(n)</a>, Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.

%H S. C. Milne, <a href="http://dx.doi.org/10.1023/A:1014865816981">Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions</a>, Ramanujan J., 6 (2002), 7-149.

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F Expansion of theta_3(z)^6.

%F a(n) = 4( Sum_{ d|n, d ==3 mod 4} d^2 - Sum_{ d|n, d ==1 mod 4} d^2 ) + 16( Sum_{ d|n, n/d ==1 mod 4} d^2 - Sum_{ d|n, n/d ==3 mod 4} d^2 ) [Jacobi]

%F a(n) = 16*A050470(n) - 4*A002173(n). - _Michel Marcus_, Dec 15 2012

%F a(n) = (12/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017

%p (sum(x^(m^2),m=-10..10))^6;

%p # Alternative:

%p A000141list := proc(len) series(JacobiTheta3(0, x)^6, x, len+1);

%p seq(coeff(%, x, j), j=0..len-1) end: A000141list(40); # _Peter Luschny_, Oct 02 2018

%t Table[SquaresR[6, n], {n, 0, 40}] (* _Ray Chandler_, Dec 06 2006 *)

%t SquaresR[6,Range[0,50]] (* _Harvey P. Dale_, Aug 26 2011 *)

%o (Haskell)

%o a000141 0 = 1

%o a000141 n = 16 * a050470 n - 4 * a002173 n

%o -- _Reinhard Zumkeller_, Jun 17 2013

%o (Sage)

%o Q = DiagonalQuadraticForm(ZZ, [1]*6)

%o Q.representation_number_list(40) # _Peter Luschny_, Jun 20 2014

%Y Row d=6 of A122141 and of A319574, 6th column of A286815.

%Y Cf. A050470, A002173.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

%E Extended by _Ray Chandler_, Nov 28 2006

%E Formula corrected by _Sean A. Irvine_, Oct 01 2009

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Last modified February 20 14:52 EST 2019. Contains 320327 sequences. (Running on oeis4.)