Fibonacci-like sequences that converge to 0

Dec 27, 10:32 AM

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

69c34c11ab0976d233bc7fee8af206d1aeb338ba

It turns out that they can, if the ratio of the second term to the first is 7249246bfa7088a808ebe568b18dd326c782dc4c, or 1-φ, approximately -0.618.

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

25e243f0a1759585683430bdfcc60e927f296abc and solving for 510842f7d1fc6a07b50c4521dd7ee6a633628623.

Here are the first few elements of the sequence:
381e6a8cefe0b5429ab77c6b4bc1ca2e90ee483c

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.

---

Comment

Commenting is closed for this article.

---