De Moivre-Type Identities for the Tetrabonacci Numbers

Abstract

It is well known that, for x² – x – 1 = 0, the two roots are $(1 \pm \sqrt 5 )/2$, and that

$${(\frac{{1 \pm \sqrt 5 }}{2})^n} = \frac{{{L_n} \pm \sqrt 5 {F_n}}}{2}$$

(1)

where L_n are the Lucas numbers and F_n the Fibonacci numbers. Identities (1) are called “de Moivre-type identities” [1].