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A005875 Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).
(Formerly M4092)
67
1, 6, 12, 8, 6, 24, 24, 0, 12, 30, 24, 24, 8, 24, 48, 0, 6, 48, 36, 24, 24, 48, 24, 0, 24, 30, 72, 32, 0, 72, 48, 0, 12, 48, 48, 48, 30, 24, 72, 0, 24, 96, 48, 24, 24, 72, 48, 0, 8, 54, 84, 48, 24, 72, 96, 0, 48, 48, 24, 72, 0, 72, 96, 0, 6, 96, 96, 24, 48, 96, 48, 0, 36, 48, 120 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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 lattice is Sum_{n>=1} (-1)^n*a(n)/sqrt(n) = -A085469. - R. J. Mathar, Apr 29 2006

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

Someone suggested that this sequence refers to the f.c.c. lattice - that is nonsense (the coordination number here, the second term in the theta series, is 6 not 12). - N. J. A. Sloane, Jan 25 2011

REFERENCES

H. Cohen, Number Theory, Vol. 1: Tools and Diophantine Equations, Springer-Verlag, 2007, p. 317.

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.

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

T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.

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

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

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').

LINKS

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

George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 2.1.

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

S. Bhargava and C. Adiga, A basic bilateral series summation formula and its applications, Integral Transforms and Special Functions, 2 (1994), 165-184.

J. M. Borwein, K-K S. Choi, On Dirichlet series for sums of squares, Raman. J. 7 (2003) 95-127

S. K. K. Choi, A. V. Kumchev and R. Osburn, On sums of three squares, arXiv:math/0502007 [math.NT], 2005.

M. Doring, J. Haidenbauer, U.-G. Meissner and A. Rusetsky, Dynamical coupled-channel approaches on a momentum lattice, arXiv preprint arXiv:1108.0676 [hep-lat], 2011.

J. A. Ewell, Recursive determination of the enumerator for sums of three squares, Internat. J. Math. and Math. Sci, 24 (2000), 529-532.

O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.

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

A. Martinez Torres, L. R. Dai, C. Koren, D. Jido and E. Oset. The KD, eta D_s interaction in finite volume and the  D_{s^*0}(2317) resonance, arXiv preprint arXiv:1109.0396 [hep-lat], 2011.

R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv preprint arXiv:1309.3705 [math.MG], 2013.

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.

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

Eric T. Mortenson, A Kronecker-type identity and the representations of a number as a sum of three squares, arXiv:1702.01627 [math.NT], 2017.

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Theta Series

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for sequences related to sums of squares

FORMULA

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

There is a classical formula (essentially due to Gauss):

For sums of 3 squares r_3(n): write (uniquely) -n=D(2^vf)^2, with D<0 fundamental discriminant, f odd, v>=-1. Then r_3(n) = 12L((D/.),0)(1-(D/2)) sum_{d | f} mu(d)(D/d)sigma(f/d).

Here mu is the Moebius function, (D/2) and (D/d) are Kronecker-Legendre symbols, sigma is the sum of divisors function, L((D/.),0)=h(D)/(w(D)/2) is the value at 0 of the L function of the quadratic character (D/.), equal to the class number h(D) divided by 2 or 3 in the special cases D=-4 and -3. - Henri Cohen (Henri.Cohen(AT)math.u-bordeaux1.fr), May 12 2010

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 function. - Michael Somos, Oct 25 2006.

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

G.f.: (Sum_{k in Z} x^k^2)^3.

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

a(n) = A004015(2*n) = A014455(2*n) = A004013(4*n) = A169783(4*n). a(4*n + 1) = 6 * A045834(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 5) = 24 * A045831(n). - Michael Somos, Jun 03 2012

a(4*n + 2) = 12 * A045828(n). - Michael Somos, Sep 03 2014

a(n) = (-1)^n * A213384(n). - Michael Somos, May 21 2015

a(n) = (6/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

EXAMPLE

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.

G.f. =  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 + ...

MAPLE

(sum(x^(m^2), m=-10..10))^3; seq(coeff(%, x, n), n=0..50);

MATHEMATICA

SquaresR[3, Range[0, 80]] (* Harvey P. Dale, Jul 21 2011 *)

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3, {q, 0, n}]; (* Michael Somos, Jun 25 2014 *)

a[ n_] := Length @ FindInstance[ n == x^2 + y^2 + z^2, {x, y, z}, Integers, 10^9]; (* Michael Somos, May 21 2015 *)

QP = QPochhammer; CoefficientList[(QP[q^2]^5/(QP[q]*QP[q^4])^2)^3 + O[q]^80, q] (* Jean-François Alcover, Nov 24 2015 *)

PROG

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

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^3, n))}; /* Michael Somos, Jun 03 2012 */

(PARI) {a(n) = my(G); if( n<0, 0, G = [ 1, 0, 0; 0, 1, 0; 0, 0, 1]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, May 21 2015 */

(Sage)

Q = DiagonalQuadraticForm(ZZ, [1]*3)

Q.representation_number_list(75) # Peter Luschny, Jun 20 2014

(MAGMA) Basis( ModularForms( Gamma1(4), 3/2), 75) [1]; /* Michael Somos, Jun 25 2014 */

(Julia) # JacobiTheta3 is defined in A000122.

A005875List(len) = JacobiTheta3(len, 3)

A005875List(75) |> println # Peter Luschny, Mar 12 2018

CROSSREFS

3rd column of A286815. - Seiichi Manyama, May 27 2017

Row d=3 of A122141.

Cf. A074590 (primitive solutions), A117609 (partial sums).

Analog for 4 squares: A000118.

x^2+y^2+k*z^2: A005875, A014455, A034933, A169783, A169784.

Cf. A004013, A004015, A008443, A045828, A045831, A045834.

Cf. A213384.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Sequence in context: A272966 A105730 A213384 * A236933 A236927 A236932

Adjacent sequences:  A005872 A005873 A005874 * A005876 A005877 A005878

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, Aug 22 2000

STATUS

approved

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Last modified May 24 09:32 EDT 2018. Contains 304517 sequences. (Running on oeis4.)