On combined 3-Fibonacci sequences

Abstract

The term ‘combined’ sequence includes any of the ‘coupled’, ‘intercalated’ and ‘pulsated’ sequences. In this paper, k = 3, so new combined 3-Fibonacci sequences, \{\alpha_n \}, \{ \beta_n \}, \{ \gamma_n \}, are introduced and the explicit formulae for their general terms are developed. That is, there are three such sequences, each with a linear recurrence relation which contains terms from the other two. In effect, each such recurrence relation is second order, with two initial terms which specify the subsequent delineation of the terms of the sequences. The initial terms are, respectively, \langle \alpha_0, \alpha_1 \rangle = \langle 2a, 2d \rangle, \langle \beta_0, \beta_1 \rangle = \langle b,e \rangle and \langle \gamma_0, \gamma_1 \rangle = \langle 2c, 2f \rangle in turn. These result in neat inter-relationships among the three sequences, which can lead to intriguing connections with known sequences, and to a surprisingly simple graphical representation of the whole process. The references include a comprehensive cover of the pertinent literature on these aspects of recursive sequences particularly during the last seventy years. A secondary goal of the paper is to put the disarray of this part of number theory into some semblance of order with a selection of representative references. This gives rise to a ‘combobulated sequence’, so-called because it restores partial order to a disarray of many papers into three classes, which are fuzzy in both their membership and non-membership because of their diverse and non-systematic derivations.