OFFSET
1,2
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.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from R. J. Mathar, Jul 16 2010
Name corrected by Andrey Zabolotskiy, Jan 29 2020
STATUS
approved