Let be a sequence with values in and let
be the first index such that divides whenever it exists (otherwise it stays undefined). Next define a sequence by the following recurrence:
for . Here, denotes Euler’s totient function. Note that the sum is extended over all with . The values of are thus defined. Moreover, the sum is finite.
What is ? As it turns out and as you might have guessed from the title is the well-known Fibonacci series.
Computational example. Assume we knew that the were the Fibonacci numbers for . Then where the sum is extended over all divisors of (i.e. the divisors of ). Thus
How does one prove that the are indeed the Fibonacci numbers? Actually, there is a nice and elementary double counting proof for this and even after scribbling it down I must confess that for me the Fibonacci numbers here somehow pop out of nowhere.