A006088 a(n) = (2^n + 2) a(n-1) (kissing number of Barnes-Wall lattice in dimension 2^n).
1, 4, 24, 240, 4320, 146880, 9694080, 1260230400, 325139443200, 167121673804800, 171466837323724800, 351507016513635840000, 1440475753672879672320000, 11803258325595576034990080000, 193408190923209108909347450880000
Offset: 0
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- 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
Crossrefs
Cf. A028362. - Paul D. Hanna, Sep 16 2009
Cf. A081845.
Programs
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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
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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
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Mathematica
RecurrenceTable[{a[0]==1, a[n]==(2^n + 2) a[n-1]}, a[n], {n, 0, 25}] (* Vincenzo Librandi, Dec 31 2015 *) -
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
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
Extensions
More terms from Emeric Deutsch, Dec 10 2004
Replaced arXiv URL with non-cached version - R. J. Mathar, Oct 23 2009