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%I #74 Sep 29 2021 09:01:34
%S 1,10,40,80,90,112,240,320,200,250,560,560,400,560,800,960,730,480,
%T 1240,1520,752,1120,1840,1600,1200,1210,2000,2240,1600,1680,2720,3200,
%U 1480,1440,3680,3040,2250,2800,3280,4160,2800,1920,4320,5040,2800,3472,5920
%N Number of ways of writing n as a sum of 5 squares.
%C 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
%C See A025429 for the number of partitions of n into five nonzero squares. - _M. F. Hasler_, May 30 2014
%C Also, theta series of lattice Z^5. - _Sean A. Irvine_, Jul 27 2020
%D E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 128.
%D J. Carlos Moreno and Samuel S. Wagstaff Jr., Sums Of Squares Of Integers, Chapman & Hall/CRC, (2006). [_Ant King_, Mar 17 2013]
%H T. D. Noe, <a href="/A000132/b000132.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. Cooper, <a href="http://dx.doi.org/10.1023/A:1021175617574">Sums of five, seven and nine squares</a>, Ramanujan J., vol 6, no. 4, (2002) 469-490.
%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 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F G.f.: (Sum_{j=-inf..+inf} x^(j^2))^5. - _R. J. Mathar_, Jul 31 2007
%F a(n) = (10/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017
%F a(n) = A000118(n) + 2*Sum_{k=1..floor(sqrt(n))} A000118(n - k^2). - _Daniel Suteu_, Aug 28 2021
%e G.f. = 1 + 10*x + 40*x^2 + 80*x^3 + 90*x^4 + 112*x^5 + 240*x^6 + ...
%t Table[SquaresR[5, n], {n, 0, 46}] (* _Ray Chandler_, Nov 28 2006 *)
%t SquaresR[5,Range[0,50]] (* _Harvey P. Dale_, Aug 26 2011 *)
%o (Sage)
%o Q = DiagonalQuadraticForm(ZZ, [1]*5)
%o Q.representation_number_list(47) # _Peter Luschny_, Jun 20 2014
%o (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
%Y 5th column of A286815. - _Seiichi Manyama_, May 27 2017
%Y Row d=5 of A122141.
%Y Cf. A000118, A025429.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Extended by _Ray Chandler_, Nov 28 2006
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