OFFSET
0,2
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.
Lomadze, G.A.: On the representations of natural numbers by sums of nine squares. Acta. Arith. 68(3), 245-253 (1994). (Russian). See Equation (3.6).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..10000
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.
M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135.
FORMULA
G.f.: theta_3(0,q)^9, where theta_3 is the 3rd Jacobi theta function. - Ilya Gutkovskiy, Jan 13 2017
a(n) = (18/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))^9;
# Alternative
A008452list := proc(len) series(JacobiTheta3(0, x)^9, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A008452list(32); # Peter Luschny, Oct 02 2018
MATHEMATICA
Table[SquaresR[9, n], {n, 0, 32}] (* Ray Chandler, Nov 28 2006 *)
PROG
(Sage)
Q = DiagonalQuadraticForm(ZZ, [1]*9)
Q.representation_number_list(37) # Peter Luschny, Jun 20 2014
(Python)
# uses Python code from A000143
from math import isqrt
def A008452(n): return A000143(n)+(sum(A000143(n-k**2) for k in range(1, isqrt(n)+1))<<1) # Chai Wah Wu, Jun 23 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Extended by Ray Chandler, Nov 28 2006
STATUS
approved