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Search: id:A004015
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| A004015 |
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Theta series of face-centered cubic (f.c.c.) lattice.
(Formerly M4821)
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+0
6
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| 1, 12, 6, 24, 12, 24, 8, 48, 6, 36, 24, 24, 24, 72, 0, 48, 12, 48, 30, 72, 24, 48, 24, 48, 8, 84, 24, 96, 48, 24, 0, 96, 6, 96, 48, 48, 36, 120, 24, 48, 24, 48, 48, 120, 24, 120, 0, 96, 24, 108, 30, 48, 72, 72, 32, 144, 0, 96, 72, 72, 48, 120, 0, 144, 12, 48, 48, 168, 48, 96
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 113.
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. 2, p. 263.
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
L. V. Woodcock, Nature, Jan 09 1997, pp. 141-143.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..9999
G. Nebe and N. J. A. Sloane, Home page for this lattice
N. J. A. Sloane, A portion of the f.c.c. lattice packing.
Index entries for sequences related to f.c.c. lattice
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FORMULA
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Expansion of phi(q^2)^3 +12*q*phi(q^2)*psi(q^4)^2 in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos Oct 25 2006
Expansion of (phi(q)^3 +phi(-q)^3)/2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos Oct 25 2006
Expansion of b(q)*phi(q^18) +c(q^3)*phi(q^2) in powers of q^3 where b(),c() are cubic AGM analog functions, and phi() is a Ramanujan theta function. - Michael Somos Oct 25 2006
Expansion of (theta_3(q)^3 + theta_4(q)^3) / 2 in powers of q^2.
G.f. is a period 1 Fourier series which satisfies f( -1 / (8 t)) = 2^(7/2) (t/i)^(3/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A004013.
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EXAMPLE
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1 + 12*q^2 + 6*q^4 + 24*q^6 + 12*q^8 + 24*q^10 + 8*q^12 + 48*q^14 + 6*q^16 + ...
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MAPLE
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maxd := 201: temp0 := trunc(evalf(sqrt(maxd)))+2: a := 0: for i from -temp0 to temp0 do a := a+q^( (i+1/2)^2): od: th2 := series(a, q, maxd); a := 0: for i from -temp0 to temp0 do a := a+q^(i^2): od: th3 := series(a, q, maxd); th4 := series(subs(q=-q, th3), q, maxd); series((1/2)*(th3^3+th4^3), q, 200);
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PROGRAM
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(PARI) {a(n)=if(n<0, 0, n*=2; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n))^3, n))} /* Michael Somos Oct 25 2006 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^5 / eta(x^2 + A)^2 / eta(x^8 + A)^2)^3 + 12 * x * eta(x^4 + A)^3 * eta(x^8 + A)^2 / eta(x^2 + A)^2, n))} /* Michael Somos May 17 2008 */
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CROSSREFS
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Cf. A005901. A055039 gives the positions of the 0's in this sequence.
A005875(2n)=a(n).
Sequence in context: A084067 A075247 A040135 this_sequence A119870 A038332 A093763
Adjacent sequences: A004012 A004013 A004014 this_sequence A004016 A004017 A004018
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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