A007727 - OEIS

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

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OFFSET

0,2

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

Table of n, a(n) for n=0..24.

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

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

MAPLE

A007727 := proc(n)

if n = 0 then

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 *)

PROG

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

CROSSREFS

Cf. A045630, A060165, A022553.

Sequence in context: A179040 A241685 A009679 * A303352 A226011 A174085

Adjacent sequences: A007724 A007725 A007726 * A007728 A007729 A007730

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

Edited by Max Alekseyev, Feb 09 2016

STATUS

approved