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arXiv · 2303.13374

Golden Ratio Base Expansions of the Logarithm and Inverse Tangent of Fibonacci and Lucas Numbers

Kunle Adegoke·Jaume Oliver Lafont
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Abstract

Let $\alpha=(1+\sqrt 5)/2$, the golden ratio, and $\beta=-1/\alpha=(1 - \sqrt 5)/2$. Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, defined by $F_n=(\alpha^n -\beta^n)/\sqrt 5$ and $L_n=\alpha^n + \beta^n$, for all non-negative integers. We derive base~$\alpha$ expansions of $\log F_n$, $\log L_n$, $\arctan\dfrac1{F_n}$ and $\arctan\dfrac1{L_n}$ for all positive integers $n$.

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Kunle Adegoke, Jaume Oliver Lafont. 2023-03-19. Golden Ratio Base Expansions of the Logarithm and Inverse Tangent of Fibonacci and Lucas Numbers. https://arxiv.org/abs/2303.13374

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