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Some trigonometric identities involving Fibonacci and Lucas numbers. (English) Zbl 1201.11023

Let \(F_n\) and \(L_n\) be the Fibonacci and Lucas numbers. The authors prove the identities \[ F_n=\frac{2^{n-1}}{n}\sqrt{\prod_{k=1}^{n-1}(1-\cos\tfrac{k\pi}{n}\cos\tfrac{3k\pi}{n})},\quad n\geq 2 \] and \[ \prod_{k=0}^{n-1}(1+4\sin^2\tfrac{k\pi}{n})=L_{2n}-2=F_{2n+2}-F_{2n-2}-2,\quad n\geq 1. \]

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
05C05 Trees
15A18 Eigenvalues, singular values, and eigenvectors

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