# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a008452 Showing 1-1 of 1 %I A008452 #44 Jun 23 2024 02:23:36 %S A008452 1,18,144,672,2034,4320,7392,12672,22608,34802,44640,60768,93984, %T A008452 125280,141120,182400,262386,317376,343536,421344,557280,665280, %U A008452 703584,800640,1068384,1256562,1234080,1421184,1851264,2034720,2057280,2338560 %N A008452 Number of ways of writing n as a sum of 9 squares. %D A008452 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121. %D A008452 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314. %D A008452 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 A008452 T. D. Noe, Table of n, a(n) for n = 0..10000 %H A008452 Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032. %H A008452 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. %H A008452 M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135. %H A008452 Index entries for sequences related to sums of squares %F A008452 G.f.: theta_3(0,q)^9, where theta_3 is the 3rd Jacobi theta function. - _Ilya Gutkovskiy_, Jan 13 2017 %F A008452 a(n) = (18/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017 %p A008452 (sum(x^(m^2),m=-10..10))^9; %p A008452 # Alternative %p A008452 A008452list := proc(len) series(JacobiTheta3(0, x)^9, x, len+1); %p A008452 seq(coeff(%, x, j), j=0..len-1) end: A008452list(32); # _Peter Luschny_, Oct 02 2018 %t A008452 Table[SquaresR[9, n], {n, 0, 32}] (* _Ray Chandler_, Nov 28 2006 *) %o A008452 (Sage) %o A008452 Q = DiagonalQuadraticForm(ZZ, [1]*9) %o A008452 Q.representation_number_list(37) # _Peter Luschny_, Jun 20 2014 %o A008452 (Python) %o A008452 # uses Python code from A000143 %o A008452 from math import isqrt %o A008452 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 A008452 Row d=9 of A122141 and of A319574, 9th column of A286815. %Y A008452 Cf. A008431. %K A008452 nonn %O A008452 0,2 %A A008452 _N. J. A. Sloane_ %E A008452 Extended by _Ray Chandler_, Nov 28 2006 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE