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

1, 12, 60, 160, 252, 312, 544, 960, 1020, 876, 1560, 2400, 2080, 2040, 3264, 4160, 4092, 3480, 4380, 7200, 6552, 4608, 8160, 10560, 8224, 7812, 10200, 13120, 12480, 10104, 14144, 19200, 16380, 11520, 17400, 24960, 18396, 16440, 24480, 27200

Offset

0,2

Comments

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

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.

Links

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

L. Carlitz, Note on sums of four and six squares, Proc. Amer. Math. Soc. 8 (1957), 120-124

S. H. Chan, An elementary proof of Jacobi's six squares theorem, Amer. Math. Monthly, 111 (2004), 806-811.

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.

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

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.

Index entries for sequences related to sums of squares

Formula

Expansion of theta_3(z)^6.

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]. [corrected by Sean A. Irvine, Oct 01 2009]

a(n) = 16*A050470(n) - 4*A002173(n). - Michel Marcus, Dec 15 2012

a(n) = (12/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Maple

(sum(x^(m^2), m=-10..10))^6;
# Alternative:
A000141list := proc(len) series(JacobiTheta3(0, x)^6, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000141list(40); # Peter Luschny, Oct 02 2018

Mathematica

Table[SquaresR[6, n], {n, 0, 40}] (* Ray Chandler, Dec 06 2006 *)
SquaresR[6, Range[0, 50]] (* Harvey P. Dale, Aug 26 2011 *)
EllipticTheta[3, 0, z]^6 + O[z]^40 // CoefficientList[#, z]& (* Jean-François Alcover, Dec 05 2019 *)

Prog

(Haskell)
a000141 0 = 1
a000141 n = 16 * a050470 n - 4 * a002173 n
-- Reinhard Zumkeller, Jun 17 2013
(Sage)
Q = DiagonalQuadraticForm(ZZ, [1]*6)
Q.representation_number_list(40) # Peter Luschny, Jun 20 2014

Crossrefs

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

Cf. A050470, A002173.

Sequence in context: A153792 A229616 A321465 * A328094 A300758 A332544

Adjacent sequences: A000138 A000139 A000140 * A000142 A000143 A000144

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Extended by Ray Chandler, Nov 28 2006

Status

approved