A031360 Number of symmetrically inequivalent coincidence rotations of index 2n-1 in lattice D_4.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

The aerated sequence 1, 0, 16, 0, 36, 0, 64, 0, 168,.. is multiplicative. - R. J. Mathar, Sep 30 2011

Some symmetrically distinct rotations generate the same coincidence site lattices, hence a(n) >= A331139(n). - Andrey Zabolotskiy, Jan 29 2020

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44. arXiv:math/0605222 [math.MG]
• Michael Baake and Peter Zeiner, Coincidences in 4 dimensions, Phil. Mag. 88 (2008), 2025-2032; arXiv:0712.0363 [math.MG]
• Philip Boyle Smith and David Tong, What Symmetries are Preserved by a Fermion Boundary State?, arXiv:2006.07369 [hep-th], 2020.
• Index entries for sequences related to D_4 lattice

Crossrefs

Cf. A031361, A331139, A331141.

Programs

Maple

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) ; # R. J. Mathar, Jul 16 2010

Mathematica

a[1]=1; a[n_ /; n >= 2 && IntegerQ[Log[2, n]]] = 0; a[p_?PrimeQ] := (p+1)^2; a[n_] := a[n] = If[Length[f = FactorInteger[n]] == 1, {p, r} = First[f]; (p+1)/(p-1)*p^(r-1)*(p^(r+1)+p^(r-1)-2), Times @@ (a /@ Power @@@ f)]; Table[a[n], {n, 1, 87, 2}] (* Jean-François Alcover, Apr 17 2013 *)

PARI

a(n,f=factor(2*n-1))=prod(i=1,#f~, my(p=f[i,1],e=f[i,2]); (p+1)/(p-1)*p^(e-1)*(p^(e+1)+p^(e-1)-2)) \\ Charles R Greathouse IV, Aug 26 2017

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

Sum_{k=1..n} a(k) ~ 1680 * Zeta(3) * n^3 / Pi^6. - Vaclav Kotesovec, Feb 07 2019

Extensions

More terms from R. J. Mathar, Jul 16 2010

Name corrected by Andrey Zabolotskiy, Jan 29 2020