The Fibonacci sequence can be defined with induction as Fn=Fn−1+Fn−2, where F1=0 and F2=1. It is also well-known that the limit of the ratio between two consecutive terms limFnFn+1 is the golden ratio ϕ.
In the generalized Fibonacci sequence below, the original Fibonacci sequence is Fn2.
Fnm=i=1∑mFn−im with Fnm=0 if n<m and Fmm=1.
Every sequence will also have a generalized golden ratio ϕm=n→∞limFnmFn+1m.
What is the limit of the generalized golden ratio below?
m→∞limϕm=m→∞limn→∞limFnmFn+1m