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A008452
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Number of ways of writing n as a sum of 9 squares.
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11
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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
(list;
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listen;
history;
text;
internal format)
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OFFSET
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0,2
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REFERENCES
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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).
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LINKS
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Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
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FORMULA
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G.f.: theta_3(0,q)^9, where theta_3 is the 3rd Jacobi theta function. - Ilya Gutkovskiy, Jan 13 2017
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MAPLE
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(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
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Table[SquaresR[9, n], {n, 0, 32}] (* Ray Chandler, Nov 28 2006 *)
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PROG
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(Sage)
Q = DiagonalQuadraticForm(ZZ, [1]*9)
(Python)
from math import isqrt
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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