关于k阶的斐波那契数和卢卡斯数的恒等式

Q2 Mathematics

Electronic Notes in Discrete Mathematics Pub Date : 2018-12-01 DOI:10.1016/j.endm.2018.11.006

Spiros D. Dafnis, Andreas N. Philippou, Ioannis E. Livieris

引用次数: 3

摘要

利用彩色平铺法导出了k阶Lucas数与Fibonacci数之间的关系式∑i=0nmi[li(k)+(m−2)Fi+1(k)] -∑j=3k(j−2)Fi−j+1(k)]=mn+1Fn+1(k)+k−2。此关系推广了A.T. Benjamin、J.J. Quinn、D. Marques和T. Edgar分别提出的著名Fibonacci-Lucas恒等式∑i=0n2iLi=2n+1Fn+1、∑i=0n3i(Li+Fi+1)=3n+1Fn+1和∑i=0nmi(Li+(m−2)Fi+1)=mn+1Fn+1。

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An identity relating Fibonacci and Lucas numbers of order k

The following relation between Fibonacci and Lucas numbers of order k, $\sum_{i = 0}^{n} m^{i} [l_{i}^{(k)} + (m - 2) F_{i + 1}^{(k)} - \sum_{j = 3}^{k} (j - 2) F_{i - j + 1}^{(k)}] = m^{n + 1} F_{n + 1}^{(k)} + k - 2,$ is derived by means of colored tiling. This relation generalizes the well-known Fibonacci-Lucas identities, $\sum_{i = 0}^{n} 2^{i} L_{i} = 2^{n + 1} F_{n + 1}, \sum_{i = 0}^{n} 3^{i} (L_{i} + F_{i + 1}) = 3^{n + 1} F_{n + 1}$ and $\sum_{i = 0}^{n} m^{i} (L_{i} + (m - 2) F_{i + 1}) = m^{n + 1} F_{n + 1}$ of A.T. Benjamin and J.J. Quinn, D. Marques, and T. Edgar, respectively.

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来源期刊

Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics

CiteScore

1.30

自引率

0.00%

发文量

期刊介绍： Electronic Notes in Discrete Mathematics is a venue for the rapid electronic publication of the proceedings of conferences, of lecture notes, monographs and other similar material for which quick publication is appropriate. Organizers of conferences whose proceedings appear in Electronic Notes in Discrete Mathematics, and authors of other material appearing as a volume in the series are allowed to make hard copies of the relevant volume for limited distribution. For example, conference proceedings may be distributed to participants at the meeting, and lecture notes can be distributed to those taking a course based on the material in the volume.