Notes & Extensions for A Remarkable Continued Fraction

Let the Fibonacci words be w(1) = 0, w(2) = 1, w(n+1) = w(n)w(n-1) considered as integers expressed in binary. It is known that for n >= 2 the numbers 0.(w) over bar (n) = w(n)/2(F)n-1 have the continued fraction [0; 2(0), 2(1), 2(1), 2(2), 2(3), 2(5), ... , 2(Fn-2)]. We provide a simple proof using Fibonacci-type recurrences of compositions of linear functions. We apply this to several related recurrences.