Congruent Products Under XOR; Fibbinary Numbers

[Ralf Stephan]

You wrote 
> {1,6,7}={3,4,7}={3,5,6}=...
> Numbers n such that: n XOR 6*n = 7*n.
> = A048715 ; Binary expansion matches ((0)*001)*(0*); or, 
> Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3). 
> ------------------------------------------------------------  
> {1,14,15}={3,9,10}={3,13,14}={5,9,12}={5,11,14}={6,11,13}=
> {7,9,14}={7,10,13}={7,11,12}={12,21,25}={12,37,41}=...
> Numbers n such that: n XOR 14*n = 15*n.
> = A048718 ; Binary expansion matches ((0)*0001)*(0*); or, 
> Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-4).

Well, a wild guess would be that this continues such that
the rhs in the definition reads 31*n and the last term in
the recurrence is f(n-5), and so on...  I can't check this
right now.


ralf