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A045663
Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement.
7
1, 2, 4, 6, 16, 30, 60, 126, 256, 504, 1020, 2046, 4080, 8190, 16380, 32730, 65536, 131070, 262080, 524286, 1048560, 2097018, 4194300, 8388606, 16776960, 33554400, 67108860, 134217216, 268435440, 536870910, 1073740740, 2147483646
OFFSET
0,2
LINKS
FORMULA
a(n) = 2*n*A000048(n) = n*A064355(n) for n > 0.
a(n) = Sum{d|n, d odd} mu(d) * 2^(n/d) for n > 0. - Andrew Howroyd, Sep 14 2019
MATHEMATICA
a[n_] := If[n==0, 1, 2n Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];
a /@ Range[0, 31] (* Jean-François Alcover, Sep 23 2019 *)
PROG
(PARI) a(n)={if(n<1, n==0, sumdiv(n, d, if(d%2, moebius(d)*2^(n/d))))} \\ Andrew Howroyd, Sep 14 2019
(Python)
from sympy import mobius, divisors
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
CROSSREFS
KEYWORD
nonn
STATUS
approved