A031360 - OEIS

%I #35 Sep 11 2020 02:22:32

%S 1,16,36,64,168,144,196,576,324,400,1024,576,960,1584,900,1024,2304,

%T 2304,1444,3136,1764,1936,6048,2304,3248,5184,2916,5184,6400,3600,

%U 3844,10752,7056,4624,9216,5184,5476,15360,9216,6400,14472,7056,11664,14400

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

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

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

%H Vaclav Kotesovec, <a href="/A031360/b031360.txt">Table of n, a(n) for n = 1..10000</a>

%H M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44. <a href="https://arxiv.org/abs/math/0605222">arXiv:math/0605222</a> [math.MG]

%H Michael Baake and Peter Zeiner, <a href="https://doi.org/10.1080/14786430701846206">Coincidences in 4 dimensions</a>, Phil. Mag. 88 (2008), 2025-2032; arXiv:<a href="https://arxiv.org/abs/0712.0363">0712.0363</a> [math.MG]

%H Philip Boyle Smith and David Tong, <a href="https://arxiv.org/abs/2006.07369">What Symmetries are Preserved by a Fermion Boundary State?</a>, arXiv:2006.07369 [hep-th], 2020.

%H <a href="/index/Da#D4">Index entries for sequences related to D_4 lattice</a>

%F 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))).

%F 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

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

%p read("transforms") : maxOrd := 120 :

%p ZetaNum := proc(p,nmax,f) local n ; L := [1,seq(0,n=2..p-1),f,seq(0,n=p+1..nmax)] ; end proc:

%p 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:

%p 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:

%p seq( Zetap[2*e+1],e=0..nops(Zetap)/2-1) ; # _R. J. Mathar_, Jul 16 2010

%t 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 *)

%o (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

%Y Cf. A031361, A331139, A331141.

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _R. J. Mathar_, Jul 16 2010

%E Name corrected by _Andrey Zabolotskiy_, Jan 29 2020