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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

Crossrefs

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

Cf. A050470, A002173.

Programs

Haskell

a000141 0 = 1
a000141 n = 16 * a050470 n - 4 * a002173 n
-- Reinhard Zumkeller, Jun 17 2013

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 *)

Python

from math import prod
from sympy import factorint
def A000141(n):
    if n == 0: return 1
    f = [(p,e,(0,1,0,-1)[p&3]) for p,e in factorint(n).items()]
    return (prod((p**(e+1<<1)-c)//(p**2-c) for p, e, c in f)<<2)-prod(((k:=p**2*c)**(e+1)-1)//(k-1) for p, e, c in f)<<2 # Chai Wah Wu, Jun 21 2024

Sage

Q = DiagonalQuadraticForm(ZZ, [1]*6)
Q.representation_number_list(40) # Peter Luschny, Jun 20 2014

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

Extensions

Extended by Ray Chandler, Nov 28 2006