|
| |
|
|
A031360
|
|
Number of coincidence site lattices of index 2n+1 in lattice D_4.
|
|
1
| |
|
|
1, 16, 36, 64, 168, 144, 196, 576, 324, 400, 1024, 576, 960, 1584, 900, 1024, 2304, 2304, 1444, 3136, 1764, 1936, 6048, 2304, 3248, 5184, 2916, 5184, 6400, 3600, 3844, 10752, 7056, 4624, 9216, 5184, 5476, 15360, 9216, 6400, 14472, 7056, 11664, 14400
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| The aerated sequence 1, 0, 16, 0, 36, 0, 64, 0, 168,.. is multiplicative. - R. J. Mathar, Sep 30 2011
|
|
|
REFERENCES
| M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.
Michael Baake and Peter Zeiner, Coincidences in 4 dimensions, arXiv:0712.0363; also http://www.math.uni-bielefeld.de/sfb701/files/preprints/sfb07084.pdf
|
|
|
LINKS
| Index entries for sequences related to D_4 lattice
|
|
|
FORMULA
| Dirichlet series for the aerated 1, 0, 16, 0, 36, 0, 64 ..: Product_{odd primes p} (1+p^(-s))*(1+p^(1-s))/((1-p^(1-s))*(1-p^(2-s))).
Dirichlet g.f. for the aerated sequence is Zeta(s) *Zeta(s-1)^2 *Zeta(s-2) / (Zeta(2*s) * Zeta(2*s-2)) *(1-2^(1-s)) *(1-2^(2-s))/ ( (1+2^(-s))*(1+2^(1-s)) ). - R. J. Mathar, Sep 30 2011
|
|
|
MAPLE
| Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 16 2010: (Start)
read("transforms") : maxOrd := 120 :
ZetaNum := proc(p, nmax, f) local n ; L := [1, seq(0, n=2..p-1), f, seq(0, n=p+1..nmax)] ; end proc:
Zeta := proc(p, nmax, f) local L, e; L := [1, seq(0, n=2..nmax)] ; for e from 1 do if p^e > nmax then break; else L := subsop(p^e=f^e, L) ; end if; end do: L ; end proc:
Zetap := [1, seq(0, n=2..maxOrd)] : for e from 3 to maxOrd do if isprime(e) then ZetaNum(e, maxOrd, 1) ; Zetap := DIRICHLET(Zetap, %) ; ZetaNum(e, maxOrd, e) ; Zetap := DIRICHLET(Zetap, %) ; Zeta(e, maxOrd, e) ; Zetap := DIRICHLET(Zetap, %) ; Zeta(e, maxOrd, e^2) ; Zetap := DIRICHLET(Zetap, %) ; end if; end do:
seq( Zetap[2*e+1], e=0..nops(Zetap)/2-1) ; (End)
|
|
|
CROSSREFS
| Sequence in context: A144548 A161753 A062312 * A125240 A050775 A022040
Adjacent sequences: A031357 A031358 A031359 * A031361 A031362 A031363
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 16 2010
|
| |
|
|
