0,2

Number of ordered triples (i,j,k) of integers such that n = i^2 + j^2 + k^2.

The Madelung Coulomb energy for alternating unit charges in the simple cubic (sodium chloride) lattice is sum(n=1,2,3,..infinity) (-1)^n*a(n)/sqrt(n) = -A085469. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 29 2006

A005875(A004215(k))=0 for k=1,2,3,... but no other elements of A005875 are zero. - Graeme McRae (g_m(AT)mcraefamily.com), Jan 15 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. T. Bateman, On the representations of a number as the sum of three squares, Trans. Amer. Math. Soc. 71 (1951), 70-101.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.

H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th ed., 1999, Chapter V.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 109.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 54.

L. Kronecker, Crelle, Vol. LVII (1860), p. 248; Werke, Vol. IV, p. 188.

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.

C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 43.

H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 338, Eq. (B').

W. Sierpinski, 1925. Teorja Liczb. pp. 1-410 (p.61).

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

S. K. K. Choi, A. V. Kumchev and R. Osburn, On sums of three squares

J. A. Ewell, Recursive Determination Of The Enumerator For Sums Of Three Squares

Hirschhorn, M. D. and Sellers, J. A., On Representations of a Number as a Sum of Three Squares, Discrete Mathematics 199 (1999), 85-101.

M. D. Hirshhorn & J. A. Sellers, On Representations Of A Number As A Sum Of Three Squares

G. Nebe and N. J. A. Sloane, Home page for this lattice

Index entries for sequences related to sums of squares

J. L. Mordell, The Representation Of Integers By Three Positive Squares

Eric Weisstein's World of Mathematics, Theta Series

n is representable as the sum of 3 squares iff n is not of the form 4^a (8k+7) (cf. A000378).

a(n) = 3*T(n) if n == 1,2,5,6 mod 8, = 2*T(n) if n == 3 mod 8, = 0 if n == 7 mod 8 and = a(n/4) if n == 0 mod 4, where T(n) = A117726(n) [from Moreno-Wagstaff].

"If 12E(n) is the number of representations of n as a sum of three squares, then E(n) = 2F(n) - G(n) where G(n) = number of classes of determinant -n, F(n) = number of uneven classes." - Dickson, quoting Kronecker. [Cf. A117726.]

a(n) = sum(d^2|n, b(n/d^2)), where b() = A074590() gives the number of primitive solutions.

Expansion of phi(q)^3 in powers of q where phi() is a Ramanujan theta functions. - Michael Somos Oct 25 2006

Euler transform of period 4 sequence [ 6, -9, 6, -3, ...]. - Michael Somos Oct 25 2006

a(8n+7)=0. a(4n)=a(n).

Order and signs are taken into account: a(1) = 6 from 1 = (+-1)^2 + 0^2 + 0^2, a(2) = 12 from 2 = (=-1)^2 + (+-1)^2 + 0^2; a(3) = 8 from 3 = (=-1)^2 + (+-1)^2 + (+-1)^2, etc.

Theta series is 1 + 6*q + 12*q^2 + 8*q^3 + 6*q^4 + 24*q^5 + 24*q^6 + 12*q^8 + 30*q^9 + 24*q^10 + ...

(sum(x^(m^2), m=-10..10))^3;

a[n_] := SumOfSquaresR[3, n]

(PARI) {a(n)=if(n<0, 0, polcoeff(sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n))^3, n))}

Cf. A074590 (primitive solutions).

A004015(n)=a(2n), A008443(n)=a(8n+3)/8, A045834(n)=a(4n+1)/6.

A004013(4n)=A004015(2n)=A014455(2n)=a(n).

Sequence in context: A029769 A074590 A105730 this_sequence A028659 A028643 A028627

Adjacent sequences: A005872 A005873 A005874 this_sequence A005876 A005877 A005878

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000