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A000141 Number of ways of writing n as a sum of 6 squares. 14
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 (list; graph; refs; listen; history; text; internal format)
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

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

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.

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]

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;

MATHEMATICA

Table[SquaresR[6, n], {n, 0, 40}] (* Ray Chandler, Dec 06 2006 *)

SquaresR[6, Range[0, 50]] (* Harvey P. Dale, Aug 26 2011 *)

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

6th column of A286815. - Seiichi Manyama, May 27 2017

Cf. A050470, A002173.

Row d=6 of A122141.

Sequence in context: A158443 A153792 A229616 * A279509 A008530 A112415

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

Formula corrected by Sean A. Irvine, Oct 01 2009

STATUS

approved

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Last modified June 22 14:21 EDT 2017. Contains 288632 sequences.