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

1, 14, 84, 280, 574, 840, 1288, 2368, 3444, 3542, 4424, 7560, 9240, 8456, 11088, 16576, 18494, 17808, 19740, 27720, 34440, 29456, 31304, 49728, 52808, 43414, 52248, 68320, 74048, 68376, 71120, 99456, 110964, 89936, 94864, 136080, 145222

Offset

0,2

References

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.

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, 2013

P. A. J. G. Chevalier, A "table of Mendeleev" for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium.

Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.

S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.

Index entries for sequences related to sums of squares

Formula

G.f.: theta_3(0,x)^7, where theta_3 is the third Jacobi theta function. - Robert Israel, Jul 16 2014

a(n) = (14/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Maple

series((sum(x^(m^2), m=-10..10))^7, x, 101);
# Alternative
#(requires at least Maple 17, and only works as long as a(n) <= 10^16 or so):
N:= 1000: # to get a(0) to a(N)
with(SignalProcessing):
A:= Vector(N+1, datatype=float[8], i-> piecewise(i=1, 1, issqr(i-1), 2, 0)):
A2:= Convolution(A, A)[1..N+1]:
A4:= Convolution(A2, A2)[1..N+1]:
A5:= Convolution(A, A4)[1..N+1];
A7:= Convolution(A2, A5)[1..N+1];
map(round, convert(A7, list)); # Robert Israel, Jul 16 2014
# Alternative
A008451list := proc(len) series(JacobiTheta3(0, x)^7, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A008451list(37); # Peter Luschny, Oct 02 2018

Mathematica

Table[SquaresR[7, n], {n, 0, 36}] (* Ray Chandler, Nov 28 2006 *)
SquaresR[7, Range[0, 50]] (* Harvey P. Dale, Aug 26 2011 *)

Prog

(Sage)
Q = DiagonalQuadraticForm(ZZ, [1]*7)
Q.representation_number_list(37) # Peter Luschny, Jun 20 2014

Crossrefs

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

Sequence in context: A166389 A085036 A107935 * A033276 A006858 A027818

Adjacent sequences: A008448 A008449 A008450 * A008452 A008453 A008454

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Extended by Ray Chandler, Nov 28 2006

Status

approved