login
Number of 2n-bead balanced binary necklaces of fundamental period 2n which are equivalent to their reverse, complement and reversed complement.
10

%I #25 Apr 15 2024 09:36:27

%S 1,1,1,1,2,3,3,7,8,14,15,31,30,63,63,123,128,255,252,511,510,1015,

%T 1023,2047,2040,4092,4095,8176,8190,16383,16365,32767,32768,65503,

%U 65535,131061,131040,262143,262143,524223,524280,1048575,1048509,2097151

%N Number of 2n-bead balanced binary necklaces of fundamental period 2n which are equivalent to their reverse, complement and reversed complement.

%H Andrew Howroyd, <a href="/A045683/b045683.txt">Table of n, a(n) for n = 0..200</a>

%H J. E. Iglesias, <a href="https://doi.org/10.1524/zkri.2006.221.4.237">Enumeration of closest-packings by the space group: a simple approach</a>, Z. Krist. 221 (2006) 237-245, Table 3.

%H <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>

%F Moebius transform of A045674. - _Andrew Howroyd_, Sep 29 2017

%F From _Andrew Howroyd_, Oct 02 2019: (Start)

%F a(n) = Sum_{d|n, d odd} mu(d) * 2^floor((n/d-1)/2) for n > 0.

%F G.f.: 1 + Sum_{k>0} mu(2*k-1)*x^(2*k-1)*(1 + x^(2*k-1))/(1 - 2*x^(4*k-2)).

%F (End)

%p A045683 := proc(p)

%p option remember ;

%p if p = 0 then

%p return 1;

%p end if;

%p a := 2^(floor((p+1)/2)-1) ;

%p for d in numtheory[divisors](p) do

%p if d >1 and type(d,'odd') then

%p a := a-procname(p/d) ;

%p end if;

%p end do:

%p a ;

%p end proc:

%p seq(A045683(p),p=0..30) ; # [Iglesias eq 12] _R. J. Mathar_, Apr 15 2024

%t b[0] = 1; b[n_] := Module[{t = 0, r = n}, While[EvenQ[r], r = Quotient[r, 2]; t += 2^(r-1)]; t + 2^Quotient[r, 2]];

%t a[0] = 1; a[n_] := DivisorSum[n, MoebiusMu[n/#]*b[#]&];

%t Table[a[n], {n, 0, 43}] (* _Jean-François Alcover_, Sep 30 2017, after _Andrew Howroyd_ *)

%o (PARI) a(n)={if(n<1, n==0, sumdiv(n, d, if(d%2, moebius(d)*2^((n/d-1)\2))))} \\ _Andrew Howroyd_, Oct 01 2019

%Y Cf. A045665, A045674, A045680, A011947 (bisection?).

%K nonn,easy

%O 0,5

%A _David W. Wilson_