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A045663 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement.

%I A045663 #19 Jul 22 2024 10:58:11
%S A045663 1,2,4,6,16,30,60,126,256,504,1020,2046,4080,8190,16380,32730,65536,
%T A045663 131070,262080,524286,1048560,2097018,4194300,8388606,16776960,
%U A045663 33554400,67108860,134217216,268435440,536870910,1073740740,2147483646
%N A045663 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement.
%H A045663 Andrew Howroyd, <a href="/A045663/b045663.txt">Table of n, a(n) for n = 0..1000</a>
%F A045663 a(n) = 2*n*A000048(n) = n*A064355(n) for n > 0.
%F A045663 a(n) = Sum{d|n, d odd} mu(d) * 2^(n/d) for n > 0. - _Andrew Howroyd_, Sep 14 2019
%t A045663 a[n_] := If[n==0, 1, 2n Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];
%t A045663 a /@ Range[0, 31] (* _Jean-François Alcover_, Sep 23 2019 *)
%o A045663 (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 A045663 (Python)
%o A045663 from sympy import mobius, divisors
%o A045663 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 A045663 Cf. A000048, A007727, A045662, A045664, A064355.
%K A045663 nonn
%O A045663 0,2
%A A045663 _David W. Wilson_