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A005875
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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)
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31
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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; internal format)
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OFFSET
| 0,2
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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=1,2,3,..infinity) (-1)^n*a(n)/sqrt(n) = -A085469. - R. J. Mathar, 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
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
| 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.
M. Doring, J. Haidenbauer, U.-G. Meissner and A. Rusetsky, Dynamical coupled-channel approaches on a momentum lattice, Arxiv preprint arXiv:1108.0676, 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.
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.
T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.
W. Sierpinski, 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').
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, 2011
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LINKS
| 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
J. L. Mordell, The Representation Of Integers By Three Positive Squares
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
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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 functions. - Michael Somos, Oct 25 2006.
Euler transform of period 4 sequence [ 6, -9, 6, -3, ...]. - Michael Somos, Oct 25 2006
G.f.: (Sum_k x^k^2)^3.
a(8*n + 7) = 0. a(4*n) = a(n).
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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.
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 + ...
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MAPLE
| (sum(x^(m^2), m=-10..10))^3;
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MATHEMATICA
| SquaresR[3, Range[0, 80]] (* From Harvey P. Dale, Jul 21 2011 *)
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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))}
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CROSSREFS
| Cf. A074590 (primitive solutions), A117609 (partial sums).
A004015(n)=a(2n), A008443(n)=a(8n+3)/8, A045834(n)=a(4n+1)/6.
A004013(4n)=A004015(2n)=A014455(2n)=a(n).
x^2+y^2+k*z^2: A005875, A014455, A034933, A169783, A169784.
Sequence in context: A029769 A074590 A105730 * A028659 A028643 A028627
Adjacent sequences: A005872 A005873 A005874 * A005876 A005877 A005878
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000
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
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