Generalized Fibonacci Numbers are Rounded Powers

Renato M. Capocelli and Paul Cull

A chapter in Applications of Fibonacci Numbers, 1990, pp 57-62 from Springer

Abstract: Abstract We show that the k th degree generalized Fibonacci number of order n ≥ 0,F n (k) can be computed by $$\begin{array}{l} F_n^{\left( k \right)} = ROUND\left\{ {\alpha _0^{\left( k \right)}{{\left( {\Phi _0^{\left( k \right)}} \right)}^n}} \right\}\\ = \left\lfloor {\frac{{\Phi _0^{\left( k \right)} - 1}}{{(k + 1)\Phi _0^{\left( k \right)} - 2k}}{{\left( {\Phi _0^{\left( k \right)}} \right)}^{n - 1}} + .5} \right\rfloor \end{array}$$ where Ф 0 (k) is the unique positive root of the characteristic polynomial for the Fibonacci numbers of degree k and ⌊x⌋ denotes the integral part of x.

Date: 1990
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DOI: 10.1007/978-94-009-1910-5_6

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