I was curious about whether Fibonacci-like sequences could converge to 0, with a proper choice of starting terms and the usual recurrence relationship:

It turns out that they can, if the ratio of the second term to the first is , or 1-φ, approximately -0.618.

I found this using the closed form for Fibonacci-like sequences

and solving for .

Here are the first few elements of the sequence:

Interestingly, the integer part are the Lucas numbers and the multiples of the square root are the Fibonacci numbers, and their ratio converges to the square root of 5, based on their closed forms.