A008451 - OEIS

%I #59 Jun 23 2024 02:21:56

%S 1,14,84,280,574,840,1288,2368,3444,3542,4424,7560,9240,8456,11088,

%T 16576,18494,17808,19740,27720,34440,29456,31304,49728,52808,43414,

%U 52248,68320,74048,68376,71120,99456,110964,89936,94864,136080,145222

%N Number of ways of writing n as a sum of 7 squares.

%D E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

%H T. D. Noe, <a href="/A008451/b008451.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 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 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. 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.: theta_3(0,x)^7, where theta_3 is the third Jacobi theta function. - _Robert Israel_, Jul 16 2014

%F a(n) = (14/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017

%p series((sum(x^(m^2),m=-10..10))^7, x, 101);

%p # Alternative

%p #(requires at least Maple 17, and only works as long as a(n) <= 10^16 or so):

%p N:= 1000: # to get a(0) to a(N)

%p with(SignalProcessing):

%p A:= Vector(N+1,datatype=float[8],i-> piecewise(i=1,1,issqr(i-1),2,0)):

%p A2:= Convolution(A,A)[1..N+1]:

%p A4:= Convolution(A2,A2)[1..N+1]:

%p A5:= Convolution(A,A4)[1..N+1];

%p A7:= Convolution(A2,A5)[1..N+1];

%p map(round,convert(A7,list)); # _Robert Israel_, Jul 16 2014

%p # Alternative

%p A008451list := proc(len) series(JacobiTheta3(0, x)^7, x, len+1);

%p seq(coeff(%,x,j), j=0..len-1) end: A008451list(37); # _Peter Luschny_, Oct 02 2018

%t Table[SquaresR[7, n], {n, 0, 36}] (* _Ray Chandler_, Nov 28 2006 *)

%t SquaresR[7,Range[0,50]] (* _Harvey P. Dale_, Aug 26 2011 *)

%o (Sage)

%o Q = DiagonalQuadraticForm(ZZ, [1]*7)

%o Q.representation_number_list(37) # _Peter Luschny_, Jun 20 2014

%o (Python)

%o # uses Python code from A000141

%o from math import isqrt

%o 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 Row d=7 of A122141 and of A319574, 7th column of A286815.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E Extended by _Ray Chandler_, Nov 28 2006