A000156 Number of ways of writing n as a sum of 24 squares.

1, 48, 1104, 16192, 170064, 1362336, 8662720, 44981376, 195082320, 721175536, 2319457632, 6631997376, 17231109824, 41469483552, 93703589760, 200343312768, 407488018512, 793229226336, 1487286966928, 2697825744960, 4744779429216

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The Carlitz paper has the wrong formula on p. 505, eq. (3). The factor in front of tau(n/2) should be -2^16 (not -2^12). The mistake appeared in the previous equation derived from eq. (2) where theta_3^(24) * 256*k^4*k'^4 was replaced by 2^8*g(q^2) which produces the factor 2^8*256 = 2^16. (One should subtract on p. 504 the second equation in the middle from the negative of the first equation. There is also a sign mistake in the sum term of the third equation from the bottom.) - Wolfdieter Lang, Sep 24 2016

References

• Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 195, eq. (15.1).
• E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 107.
• G. H. Hardy, Ramanujan, 1940, Cambridge, reprinted with additional corrections and comments by AMS Chelsea Publishing, 1999, 2002, Providence, Rhode Island, ch. IX., pp. 153-155.
• 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
• L. Carlitz, On the representation of an integer as the sum of twenty-four squares, Indagationes Mathematicae (Proceedings), 58 (1955) 504-506.
• H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
• 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=24 of A122141 and of A319574, 24th column of A286815.

Programs

Maple

(sum(x^(m^2),m=-10..10))^24; seq(coeff(%,x,n), n=0..30);
# Alternative:
A000156list := proc(len) series(JacobiTheta3(0, x)^24, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000156list(21); # Peter Luschny, Oct 02 2018

Mathematica

Table[SquaresR[24, n], {n, 0, 20}] (* Ray Chandler, Nov 28 2006 *)

PARI

first(n)=my(x='x); x+=O(x^(n+1)); Vec((2*sum(k=1,sqrtint(n),x^k^2) + 1)^24) \\ Charles R Greathouse IV, Jul 29 2016

Formula

From Wolfdieter Lang, Sep 24 2016: (Start)

For n >= 1: a(n) = (16*sigma^*{11} - 128*(512*tau(n/2) + (-1)^n*259*tau(n)))/691, with sigma^*{11} = sigma_{11}^{e}(n) - sigma_{11}^{o}(n) if n even and sigma_{11}(n) otherwise. Here sigma_{11}(n) = A013959(n) and 0 if n is not an integer, sigma_{11}^{e}(n) and sigma_{11}^{o}(n) are the sums of the 11th power of the odd and even positive divisors of n, respectively. Ramanujan's tau(n) = A000594(n) and 0 if n is not an integer. This is from Hardy, ch. IX., p. 155, eqs. (9.17.1) and (9.17.2), and p.142 for the definition of sigma^*_{nu}(n). See also the Ash and Gross reference.

Another version, see the corrected Carlitz reference:

a(n) = (2^4*(sigma_{11}(n)- 2*sigma_{11}(n/2) + 2^{12}*sigma_{11}(n/4)) - 2^7*259*(-1)^n*tau(n) - 2^16*tau(n/2))/691, n >= 1.

(End)

a(n) = (48/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