A045663 - OEIS

%I #19 Jul 22 2024 10:58:11

%S 1,2,4,6,16,30,60,126,256,504,1020,2046,4080,8190,16380,32730,65536,

%T 131070,262080,524286,1048560,2097018,4194300,8388606,16776960,

%U 33554400,67108860,134217216,268435440,536870910,1073740740,2147483646

%N Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement.

%H Andrew Howroyd, <a href="/A045663/b045663.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = 2*n*A000048(n) = n*A064355(n) for n > 0.

%F a(n) = Sum{d|n, d odd} mu(d) * 2^(n/d) for n > 0. - _Andrew Howroyd_, Sep 14 2019

%t a[n_] := If[n==0, 1, 2n Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];

%t a /@ Range[0, 31] (* _Jean-François Alcover_, Sep 23 2019 *)

%o (PARI) a(n)={if(n<1, n==0, sumdiv(n, d, if(d%2, moebius(d)*2^(n/d))))} \\ _Andrew Howroyd_, Sep 14 2019

%o (Python)

%o from sympy import mobius, divisors

%o def A045663(n): return sum(mobius(d)<<n//d for d in divisors(n>>(~n&n-1).bit_length(),generator=True)) if n else 1 # _Chai Wah Wu_, Jul 22 2024

%Y Cf. A000048, A007727, A045662, A045664, A064355.

%K nonn

%O 0,2

%A _David W. Wilson_