AN INTERESTING LINEAR RECURRENCE RELATIONSHIP

Abstract

Leonardo Pisano, originally from Pisa, wrote a book entitled Liber Abaci in 1202. Amongst many great attributions, one contribution, marked insignificant for many years, was an unanswered theoretical question. This question introduced a sequence now known to the world as the Fibonacci sequence. In the Fibonacci Sequence, Every number in the sequence is the sum of the two preceding numbers, such that , {1,1,2,3,5,8,13…}. In this research, we studied a second order recurrence relation with the constant coefficients of 1, and 2. The equation is defined as with the seed values of and . While this sequence shares several properties with the Fibonacci sequence, it also manifests exclusive properties of its own. We have found closed form of our relationship, and another unique equation for our relation . We investigated for our sequence the limit of as approaches infinity using our closed-form equation. We found a relationship for the sum of our relationship. And used different seed values including using the square of the first seed value as the second seed value in attempts of finding any unique relationship. Finally, we examined the Pisano periods and modular patterns for our different seed values and the Fibonacci sequence.