A008452 Number of ways of writing n as a sum of 9 squares.
1, 18, 144, 672, 2034, 4320, 7392, 12672, 22608, 34802, 44640, 60768, 93984, 125280, 141120, 182400, 262386, 317376, 343536, 421344, 557280, 665280, 703584, 800640, 1068384, 1256562, 1234080, 1421184, 1851264, 2034720, 2057280, 2338560
Offset: 0
Keywords
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.
- Index entries for sequences related to sums of squares
Programs
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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
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Mathematica
Table[SquaresR[9, n], {n, 0, 32}] (* Ray Chandler, Nov 28 2006 *) -
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
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Sage
Q = DiagonalQuadraticForm(ZZ, [1]*9) Q.representation_number_list(37) # Peter Luschny, Jun 20 2014
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
Extensions
Extended by Ray Chandler, Nov 28 2006