Generalization is my frenemy

Consider the generalization of the Fibonacci sequence: define the \(n\)-bonacci sequence such that its term is sum of the previous \(n\) terms with initial terms: \(a_{0,n} = a_{1,n} = a_{2,n} = \ldots = a_{n-2,n} = 0, a_{n-1,n} = 1 \).

Now we denote the \(m^\text{th} \) term of the \(n\)-bonacci sequence as \(W_{m,n} \). Calculate the limit below.

\[ \large \lim_{n \to \infty} \lim_{m \to \infty} \frac { W_{m+1,n} }{W_{m,n}} \]

Details and Assumptions:

  • As an explicit example: if \(n=2\), we will get a Fibonacci sequence, which means the limit is \( \phi = \frac { 1 + \sqrt 5}{2} \).

  • You do not need computational aids to solve this problem.

Inspirations: Problem 1 and Problem 2.

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