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
Leonard Carlitz,Note on sums of four and six squares, Proc. Amer. Math. Soc. 8 (1957), 120-124
Song Heng Chan,An elementary proof of Jacobi's six squares theorem, Amer. Math. Monthly, 111 (2004), 806-811.
Heng Huat Chan and Christian 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.
Stephen C. Milne,Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
Simon Plouffe,Numbers in the base e^Pi, arXiv:2509.15609 [math.NT], 2025. See p. 9/24, marked 6.
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 bySean A. Irvine, Oct 01 2009]
a(n) = (12/n)*Sum_{k=1..n}A186690(k)*a(n-k), a(0) = 1. -Seiichi Manyama, May 27 2017
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = Pi^(3/2) / Gamma(3/4)^6 =A388075. -Simon Plouffe, Sep 14 2025
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
(SageMath)
Q = DiagonalQuadraticForm(ZZ, [1]*6)
Q.representation_number_list(40) #Peter Luschny, Jun 20 2014
(Python)
from math import prod
from sympy import factorint
defA000141(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
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Extended byRay Chandler, Nov 28 2006
STATUS
approved