A006088 a(n) = (2^n + 2) a(n-1) (kissing number of Barnes-Wall lattice in dimension 2^n). (Formerly M3606)

1, 4, 24, 240, 4320, 146880, 9694080, 1260230400, 325139443200, 167121673804800, 171466837323724800, 351507016513635840000, 1440475753672879672320000, 11803258325595576034990080000, 193408190923209108909347450880000

Offset

0,2

References

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

Links

Table of n, a(n) for n=0..14.

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 151.

Robert L. Griess Jr., Pieces of 2^d: Existence and uniqueness for Barnes-Wall and Ypsilanti lattices, arXiv:math/0403480 [math.GR], Mar 28 2004. See Proposition 8.9.

J. Leech, Some sphere packings in higher space, Canad. J. Math., 16 (1964), 657-682.

J. Leech & N. J. A. Sloane, Correspondence, 1975

C. Musès, The dimensional family approach in (hyper)sphere packing..., Applied Math. Computation 88 (1997), pp. 1-26, see p. 22.

G. Nebe and N. J. A. Sloane, Table of highest kissing numbers known

Index entries for sequences related to Barnes-Wall lattices

Formula

a(n) = (2+2)(2+4)(2+8)(2+16)...(2+2^n).

From Paul D. Hanna, Sep 16 2009: (Start)

G.f.: Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=1..n+1} (1-2^k*x)];

contrast with:

1 = Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=1..n+1} (1+2^k*x)]. (End)

a(n) ~ c * 2^(n*(n+1)/2), where c = A081845. - Vaclav Kotesovec, Dec 31 2015

Maple

a[0]:=1: for n from 1 to 16 do a[n]:=(2^n+2)*a[n-1] od: seq(a[n], n=0..16); # Emeric Deutsch, Dec 10 2004

Mathematica

RecurrenceTable[{a[0]==1, a[n]==(2^n + 2) a[n-1]}, a[n], {n, 0, 25}] (* Vincenzo Librandi, Dec 31 2015 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, 2^(m*(m+1)/2)*x^m/prod(k=1, m+1, 1-2^k*x+x*O(x^n))), n)} \\ Paul D. Hanna, Sep 16 2009
(PARI) a(n) = prod(k=1, n, 2+2^k); \\ Michel Marcus, Jan 01 2016
(Magma) I:=[4]; [1] cat [n le 1 select I[n] else (2^n + 2)*Self(n-1): n in [1..20]]; // Vincenzo Librandi, Dec 31 2015

Crossrefs

Cf. A028362. - Paul D. Hanna, Sep 16 2009

Cf. A081845.

Sequence in context: A052718 A061640 A126391 * A325963 A141013 A330469

Adjacent sequences: A006085 A006086 A006087 * A006089 A006090 A006091

Keyword

nonn,easy

Author

N. J. A. Sloane, John Leech

Extensions

More terms from Emeric Deutsch, Dec 10 2004

Replaced arXiv URL with non-cached version - R. J. Mathar, Oct 23 2009

Status

approved