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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Crossrefs

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

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

Programs

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 *)

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

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

Extensions

Extended by Ray Chandler, Nov 28 2006