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Maria Adam 1 and Nicholas Assimakis 1, 2
Academic Editor:H. Deng and Academic Editor:E. Gyori and Academic Editor:B. Zhou
1, Department of Computer Science and Biomedical Informatics, University of Thessaly, 2-4 Papasiopoulou street, 35100 Lamia, Greece
2, Department of Electronic Engineering, Technological Educational Institute of Central Greece, 3rd km Old National Road Lamia-Athens, 35100 Lamia, Greece
Received 24 November 2013; Accepted 25 February 2014; 9 April 2014
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
It is well-known that the Fibonacci sequence, the Lucas sequence, the Padovan sequence, the Perrin sequence, the tribonacci sequence, and the tetranacci sequence are very prominent examples of recursive sequences, which are defined as follows.
The Fibonacci numbers 1,1 , 2,3 , 5,8 , 13 , ... are derived by the recurrence relation f n = f n - 1 + f n - 2 , n ...5; 3 , with f 1 = f 2 = 1 , [1], [2, A000045].
The Lucas numbers 2,1 , 3,4 , 7,11,18,29 , ... are derived by the recurrence relation [cursive l] n = [cursive l] n - 1 + [cursive l] n - 2 , n ...5; 3 , with [cursive l] 1 = 2 , and [cursive l] 2 = 1 , [1], [2, A000032].
The Padovan numbers 1,0 , 0,1 , 0,1 , 1,1 , 2,2 , 3,4 , 5,7 , 9,12 , ... are derived by the recurrence relation a n = a n - 2 + a n - 3 , n ...5; 4 , with a 1 = 1 , a 2 = a 3 = 0 , [2, A000931].
The Perrin numbers 3,0 , 2,3 , 2,5 , 5,7 , 10,12,17 , ... are derived by the recurrence relation p n = p n - 2 + p n - 3 , n ...5; 4 , with p 1 = 3 , p 2 = 0 , and p 3 = 2 , [2, A001608].
Both Fibonacci and Lucas numbers as well as both Padovan and Perrin numbers satisfy the same recurrence relation with...