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Three series for the generalized golden mean. (English) Zbl 1384.11025

Summary: As is well-known, the ratio of adjacent Fibonacci numbers tends to \( \varphi = (1 + \sqrt{5} )/2\), and the ratio of adjacent Tribonacci numbers (where each term is the sum of the three preceding numbers) tends to the real root \(\eta\) of \(X^3 - X^2 - X - 1 = 0\). Letting \(\alpha_n\) denote the corresponding ratio for the generalized Fibonacci numbers, where each term is the sum of the n preceding, we obtain rapidly converging series for \(\alpha_n\), \(1/\alpha_n\), and \(1/(2 - \alpha_n)\).

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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