A "Lucas Inversion Formula" for "Number of aperiodic binary necklaces with no subsequence 00"

[Max]

Creighton Dement wrote:

> The formula given for A006206 is 1/n * sum over d divides n of {mu(n/d)
> * Lucas_d} 
> 
> Ex. A006206(5) = (1/5)*(mu(5)*Lucas(1) + mu(1)*Lucas(5) ) = (1/5)*(-1 +
> 11) = 2. 
> 
> However, a closer inspection of the "Lucas Strings" leads me to
> conjecture:
> 
> Lucas(n) =  sum over d divides n of d*A006206(d)
> 
> Ex. Lucas(5) =  11 = 1*1 + 5*2  = 1*A006206(1) + 5*A006206(5) 
> Lucas(18) = 5778 =  1*1 + 2*1 + 3*1 + 6*2 +  9*8 + 18*316 = 
> 1*A006206(1) + 2*A006206(2)  + 3*A006206(3) + 6*A006206(6) +
> 9*A006206(9)

It's direct consequence of Möbius Inversion Formula http://mathworld.wolfram.com/MoebiusInversionFormula.html
applied to the functions f(n)=n*A006206(n) and g(n)=Lucas(n).

Max