Widely Generalized Fibonacci Numbers Induced by Systematical Generation of Widely Generalized Morse Codes and the Matrix Representations

Abstract

In our previous paper[17], we naturally generalized the Morse code and we found the associative generalized Fibonacci sequences. Further we studied in[18]the matrix representation of these generalized sequences. In this paper, we introduce a new code which is developed by our preceding studies of the generalized Morse code. Moreover, we examine an efficient algorithm for generating codewords of the new code systematically and show that the number of codeword of equal lengths gives more widely generalized Fibonacci sequences. Subsequently we also introduce the associated widely generalized Lucas numbers and we study the direct representation of these n-th terms of the newly generalized Fibonacci and Lucas sequences by making use of matrices. Furthermore, we study some extended properities concerning these widely generalized sequences.