A000152 Number of ways of writing n as a sum of 16 squares.

1, 32, 480, 4480, 29152, 140736, 525952, 1580800, 3994080, 8945824, 18626112, 36714624, 67978880, 118156480, 197120256, 321692928, 509145568, 772845120, 1143441760, 1681379200, 2428524096, 3392205824, 4658843520, 6411152640

Offset

0,2

References

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

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 107.

Links

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

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,q)^16, where theta_3 is the 3rd Jacobi theta function. - Ilya Gutkovskiy, Jan 13 2017

a(n) = (32/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Maple

(sum(x^(m^2), m=-10..10))^16;
# Alternative:
A000152list := proc(len) series(JacobiTheta3(0, x)^16, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000152list(24); # Peter Luschny, Oct 02 2018

Mathematica

Table[SquaresR[16, n], {n, 0, 23}] (* Ray Chandler, Nov 28 2006 *)
CoefficientList[EllipticTheta[3, 0, x]^16 + O[x]^24, x] (* Jean-François Alcover, Jul 06 2017 *)

Prog

(PARI) first(n)=my(x='x); x+=O(x^(n+1)); Vec((2*sum(k=1, sqrtint(n), x^k^2) + 1)^16) \\ Charles R Greathouse IV, Jul 29 2016

Crossrefs

Row d=16 of A122141 and of A319574, 16th column of A286815.

Cf. A022047(n) = A000152(2*n).

Sequence in context: A125467 A282525 A250319 * A319307 A022069 A250560

Adjacent sequences: A000149 A000150 A000151 * A000153 A000154 A000155

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Extended by Ray Chandler, Nov 28 2006

Status

approved