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A008451 Number of ways of writing n as a sum of 7 squares.

%I A008451 #59 Jun 23 2024 02:21:56
%S A008451 1,14,84,280,574,840,1288,2368,3444,3542,4424,7560,9240,8456,11088,
%T A008451 16576,18494,17808,19740,27720,34440,29456,31304,49728,52808,43414,
%U A008451 52248,68320,74048,68376,71120,99456,110964,89936,94864,136080,145222
%N A008451 Number of ways of writing n as a sum of 7 squares.
%D A008451 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
%D A008451 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.
%H A008451 T. D. Noe, <a href="/A008451/b008451.txt">Table of n, a(n) for n = 0..10000</a>
%H A008451 Philippe A. J. G. Chevalier, <a href="https://www.researchgate.net/publication/236594822_On_the_discrete_geometry_of_physical_quantities">On the discrete geometry of physical quantities</a>, 2013
%H A008451 P. A. J. G. Chevalier, <a href="http://www.researchgate.net/profile/Philippe_Chevalier2/publication/262067273_A_table_of_Mendeleev_for_physical_quantities/links/0c9605368f6d191478000000.pdf">A "table of Mendeleev" for physical quantities?</a>, Slides from a talk, May 14 2014, Leuven, Belgium.
%H A008451 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 A008451 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 A008451 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A008451 G.f.: theta_3(0,x)^7, where theta_3 is the third Jacobi theta function. - _Robert Israel_, Jul 16 2014
%F A008451 a(n) = (14/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017
%p A008451 series((sum(x^(m^2),m=-10..10))^7, x, 101);
%p A008451 # Alternative
%p A008451 #(requires at least Maple 17, and only works as long as a(n) <= 10^16 or so):
%p A008451 N:= 1000: # to get a(0) to a(N)
%p A008451 with(SignalProcessing):
%p A008451 A:= Vector(N+1,datatype=float[8],i-> piecewise(i=1,1,issqr(i-1),2,0)):
%p A008451 A2:= Convolution(A,A)[1..N+1]:
%p A008451 A4:= Convolution(A2,A2)[1..N+1]:
%p A008451 A5:= Convolution(A,A4)[1..N+1];
%p A008451 A7:= Convolution(A2,A5)[1..N+1];
%p A008451 map(round,convert(A7,list)); # _Robert Israel_, Jul 16 2014
%p A008451 # Alternative
%p A008451 A008451list := proc(len) series(JacobiTheta3(0, x)^7, x, len+1);
%p A008451 seq(coeff(%,x,j), j=0..len-1) end: A008451list(37); # _Peter Luschny_, Oct 02 2018
%t A008451 Table[SquaresR[7, n], {n, 0, 36}] (* _Ray Chandler_, Nov 28 2006 *)
%t A008451 SquaresR[7,Range[0,50]] (* _Harvey P. Dale_, Aug 26 2011 *)
%o A008451 (Sage)
%o A008451 Q = DiagonalQuadraticForm(ZZ, [1]*7)
%o A008451 Q.representation_number_list(37) # _Peter Luschny_, Jun 20 2014
%o A008451 (Python)
%o A008451 # uses Python code from A000141
%o A008451 from math import isqrt
%o A008451 def A008451(n): return A000141(n)+(sum(A000141(n-k**2) for k in range(1,isqrt(n)+1))<<1) # _Chai Wah Wu_, Jun 23 2024
%Y A008451 Row d=7 of A122141 and of A319574, 7th column of A286815.
%K A008451 nonn
%O A008451 0,2
%A A008451 _N. J. A. Sloane_
%E A008451 Extended by _Ray Chandler_, Nov 28 2006