Search: id:A004015
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%I A004015 M4821
%S A004015 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,
%T A004015 84,24,96,48,24,0,96,6,96,48,48,36,120,24,48,24,48,48,120,24,120,0,96,
%U A004015 24,108,30,48,72,72,32,144,0,96,72,72,48,120,0,144,12,48,48,168,48,96
%N A004015 Theta series of face-centered cubic (f.c.c.) lattice.
%D A004015 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A004015 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 113.
%D A004015 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.
%D A004015 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.
%D A004015 L. V. Woodcock, Nature, Jan 09 1997, pp. 141-143.
%H A004015 N. J. A. Sloane, <a href="b004015.txt">Table of n, a(n) for n = 0..9999</
               a>
%H A004015 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               lattices/D3.html">Home page for this lattice</a>
%H A004015 N. J. A. Sloane, <a href="a000292a.jpg">A portion of the f.c.c. lattice 
               packing.</a>
%H A004015 <a href="Sindx_Fa.html#fcc">Index entries for sequences related to f.c.c. 
               lattice</a>
%H A004015 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ThetaSeries.html">Theta Series</a>
%F A004015 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
%F A004015 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
%F A004015 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
%F A004015 Expansion of (theta_3(q)^3 + theta_4(q)^3) / 2 in powers of q^2.
%F A004015 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.
%e A004015 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 
               + ...
%p A004015 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);
%o A004015 (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 */
%o A004015 (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 */
%Y A004015 Cf. A005901. A055039 gives the positions of the 0's in this sequence.
%Y A004015 A005875(2n)=a(n).
%Y A004015 Sequence in context: A084067 A075247 A040135 this_sequence A119870 A038332 
               A093763
%Y A004015 Adjacent sequences: A004012 A004013 A004014 this_sequence A004016 A004017 
               A004018
%K A004015 nonn,easy,nice
%O A004015 0,2
%A A004015 N. J. A. Sloane (njas(AT)research.att.com).

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