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A000156 Number of ways of writing n as a sum of 24 squares. 2
1, 48, 1104, 16192, 170064, 1362336, 8662720, 44981376, 195082320, 721175536, 2319457632, 6631997376, 17231109824, 41469483552, 93703589760, 200343312768, 407488018512, 793229226336, 1487286966928, 2697825744960, 4744779429216 (list; graph; refs; listen; history; text; internal format)



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


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.


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


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.



(sum(x^(m^2), m=-10..10))^24; seq(coeff(%, x, n), n=0..30);


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


Sequence in context: A160068 A229387 A010839 * A022077 A010964 A035719

Adjacent sequences:  A000153 A000154 A000155 * A000157 A000158 A000159




N. J. A. Sloane


Extended by Ray Chandler, Nov 28 2006



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Last modified March 28 11:45 EDT 2017. Contains 284186 sequences.