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arXiv · 1401.6200

Three Series for the Generalized Golden Mean

Kevin Hare·Helmut Prodinger·Jeffrey Shallit
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Abstract

As is well-known, the ratio of adjacent Fibonacci numbers tends to phi = (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)).

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Kevin Hare, Helmut Prodinger, Jeffrey Shallit. 2014-01-23. Three Series for the Generalized Golden Mean. https://arxiv.org/abs/1401.6200

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