Proof of the Tojaaldi sequence conjectures

Heuristically, the base b, size a Tojaaldi sequence of size k, T-k((a, b)), is the sequence of initial digits of the (k+1)-digit Generalized Fibonaaci numbers, defined by F-0((a)) = 0, F-1((a)) = 1, F-n((a)) = aF(n-1)((a)) + F-n-2((a)), n >= 2. For example, T-2((1,10)) = < 1, 2, 3, 6, 9 > corresponding to the initial digits of the three-digit Fibonacci numbers, 144, 233, 377, 610, 987. In [1] we showed that (eventually) there are at most b Tojaaldi sequences and conjectured that there are exactly b Tojaaldi sequences. Based on computer studies we also conjectued that the Tojaaldi sequences are Benford distributed. We prove these two conjectures