A007727 Number of 2n-bead black-white strings with n black beads and fundamental period 2n.

1, 2, 4, 18, 64, 250, 900, 3430, 12800, 48600, 184500, 705430, 2703168, 10400598, 40113164, 155117250, 601067520, 2333606218, 9075085776, 35345263798, 137846344000, 538257870990, 2104098258284, 8233430727598, 32247600966144

Offset: 0

Author

Doug Bowman, bowman(AT)math.uiuc.edu

Keywords

• nonn

Comments

For n>0, a(n) is divisible by n^2 (cf. A268619) and 6*a(n) is divisible by n^3 (cf. A268592). - Max Alekseyev, Feb 07 2016

Links

• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Crossrefs

Cf. A045630, A060165, A022553.

Programs

Maple

A007727 := proc(n)
    if n = 0 then
        1;
    else
        add(numtheory[mobius](n/d)*binomial(2*d,d), d =numtheory[divisors](n)) ;
    end if ;
end proc:
seq(A007727(n),n=0..10) ; # R. J. Mathar, Nov 10 2021

Mathematica

a[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 05 2023 *)

PARI

{ a(n) = if(n>0,sumdiv(n, d, moebius(n/d)*binomial(2*d, d)),0); }

Formula

For n>0, a(n) = Sum_{d|n} A008683(n/d)*A000984(d).

For n>0, a(n) = 2 * A045630(n).

a(0)=1, a(n) = n * A060165(n) = 2n * A022553(n). - Ralf Stephan, Sep 01 2003

Extensions

Edited by Max Alekseyev, Feb 09 2016