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A000132
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Number of ways of writing n as a sum of 5 squares.
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15
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1, 10, 40, 80, 90, 112, 240, 320, 200, 250, 560, 560, 400, 560, 800, 960, 730, 480, 1240, 1520, 752, 1120, 1840, 1600, 1200, 1210, 2000, 2240, 1600, 1680, 2720, 3200, 1480, 1440, 3680, 3040, 2250, 2800, 3280, 4160, 2800, 1920, 4320, 5040, 2800, 3472, 5920
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OFFSET
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0,2
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COMMENTS
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The units digit of a(n) is 2 if n=5*t^2 for some natural number t, and 0 otherwise. See Moreno & Wagstaff, p. 258, exercise 2. - Ant King, Mar 17 2013
See A025429 for the number of partitions of n into five nonzero squares. - M. F. Hasler, May 30 2014
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REFERENCES
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E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 128.
J. Carlos Moreno and Samuel S. Wagstaff Jr., Sums Of Squares Of Integers, Chapman & Hall/CRC, (2006). [Ant King, Mar 17 2013]
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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.: (Sum_{j=-inf..+inf} x^(j^2))^5. - R. J. Mathar, Jul 31 2007
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EXAMPLE
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G.f. = 1 + 10*x + 40*x^2 + 80*x^3 + 90*x^4 + 112*x^5 + 240*x^6 + ...
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MATHEMATICA
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Table[SquaresR[5, n], {n, 0, 46}] (* Ray Chandler, Nov 28 2006 *)
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PROG
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(Sage)
Q = DiagonalQuadraticForm(ZZ, [1]*5)
(PARI) a(n, k=5) = if(n==0, return(1)); if(k <= 0, return(0)); if(k==1, return(issquare(n))); my(count = 0); for(v = 0, sqrtint(n), count += (2 - (v == 0))*if(k > 2, a(n - v^2, k-1), issquare(n - v^2) * (2 - (n - v^2 == 0)))); count; \\ Daniel Suteu, Aug 28 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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