A008452 - OEIS

%I #44 Jun 23 2024 02:23:36

%S 1,18,144,672,2034,4320,7392,12672,22608,34802,44640,60768,93984,

%T 125280,141120,182400,262386,317376,343536,421344,557280,665280,

%U 703584,800640,1068384,1256562,1234080,1421184,1851264,2034720,2057280,2338560

%N Number of ways of writing n as a sum of 9 squares.

%D E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

%D 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).

%H T. D. Noe, <a href="/A008452/b008452.txt">Table of n, a(n) for n = 0..10000</a>

%H Shi-Chao Chen, <a href="http://dx.doi.org/10.1016/j.jnt.2010.01.011">Congruences for rs(n)</a>, Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.

%H S. C. Milne, <a href="http://dx.doi.org/10.1023/A:1014865816981">Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions</a>, Ramanujan J., 6 (2002), 7-149.

%H M. Peters, <a href="http://journals.impan.gov.pl/aa/Inf/102-2-2.html">Sums of nine squares</a>, Acta Arith., 102 (2002), 131-135.

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F G.f.: theta_3(0,q)^9, where theta_3 is the 3rd Jacobi theta function. - _Ilya Gutkovskiy_, Jan 13 2017

%F a(n) = (18/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017

%p (sum(x^(m^2),m=-10..10))^9;

%p # Alternative

%p A008452list := proc(len) series(JacobiTheta3(0, x)^9, x, len+1);

%p seq(coeff(%, x, j), j=0..len-1) end: A008452list(32); # _Peter Luschny_, Oct 02 2018

%t Table[SquaresR[9, n], {n, 0, 32}] (* _Ray Chandler_, Nov 28 2006 *)

%o (Sage)

%o Q = DiagonalQuadraticForm(ZZ, [1]*9)

%o Q.representation_number_list(37) # _Peter Luschny_, Jun 20 2014

%o (Python)

%o # uses Python code from A000143

%o from math import isqrt

%o 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

%Y Row d=9 of A122141 and of A319574, 9th column of A286815.

%Y Cf. A008431.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E Extended by _Ray Chandler_, Nov 28 2006