Incomplete Fibonacci and Lucas numbers
type: Item
instance of: scholarly article
title: Incomplete Fibonacci and Lucas numbers
zbMATH Open document ID: 0855.11007
DOI: 10.1007/BF02845088
author: Piero Filipponi
publication date: 1996-08-20T00:00:00Z
review text: It is well known that the Fibonacci numbers \(F_n\) and the Lucas numbers \(L_n\) can be written as \[ \begin{aligned} F_n &= \sum^k_{i=0} {{n-1-i} \choose i}, \qquad \lfloor (n- 1)/2 \rfloor\leq k\leq n-1, \tag{1}\\ L_n &= \sum^k_{i=0} {n\over {n-i}} {{n-i} \choose i}, \qquad \lfloor n/2 \rfloor \leq k\leq n-1. \tag{2} \end{aligned} \] The extended Fibonacci numbers \(G_n (k)\) and the extended Lucas numbers \(H_n (k)\) arise from (1) and (2) with \(k> n-1\), see the author, \textit{O. Brugia}, and \textit{A. F. Horadam} [Int. J. Math. Educ. Sci. Technol. 24, No. 1, 9-21 (1993; Zbl 0773.11014)] and the author, \textit{R. Meniocci}, and \textit{A. F. Horadam} [Fibonacci Q. 32, No. 5, 455-464 (1994; Zbl 0834.11008)]. In this paper the author studies the incomplete Fibonacci numbers \(F_n (k)\) arising from (1) with \(0\leq k\leq \lfloor (n- 1)/2 \rfloor\) and the incomplete Lucas numbers \(L_n (k)\) arising from (2) with \(0\leq k\leq \lfloor n/2 \rfloor\). The study of the congruence properties of \(L_n (k)\) leads to a new characterization of prime numbers.
reviewed by: Pentti Haukkanen
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11A41
zbMATH DE Number: 896444
zbMATH Keywords: second-order recurrence relation
zbMATH Keywords: incomplete Fibonacci numbers
zbMATH Keywords: incomplete Lucas numbers
zbMATH Keywords: congruence properties
zbMATH Keywords: characterization of prime numbers
MaRDI profile type: Publication
cites work: Q4304073
cites work: Q3834120
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cites work: A note on the improper use of a formula for Fibonacci numbers
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cites work: Uniform Huffman Sequences do not exist
published in: Rendiconti del Circolo Matematico di Palermo
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description: scientific article; zbMATH DE number 896444