The Fibonacci Sequence: From the Rabbit to Horadam

Abstract

The Fibonacci sequence is perhaps one of the most-well known concepts in the world of Mathematics. It is closely associated with another widely referenced mathematical formula known as the Golden Ratio. The purpose of this project is to lay out a foundation for these concepts, encompassing their origins, history, and some of their applications. The initial focal point presents a formal definition and introduces some basic formulas, which are used to expand upon the starting points and branch out into related topics. A link between the Fibonacci sequence and the Golden Ratio is established, including explanations with mathematical proofs where required, as well as value tables to corroborate statements. Some salient characteristics of the sequence are highlighted, going over the main aspects of related sequences such as NegaFibonacci and the Lucas numbers, while also describing noteworthy patterns in music. Notwithstanding, the research is chiefly focused on analyzing sequences using the Pisano Period. To this end, Python code is employed to write a simple program, in order to quickly generate sequences and help out with calculations. Using 2, 3, 4, and 5 with their respective squares as the starting seeds, Fibonacci sequences are generated alongside a staircase plot for each case. The results show some similarities in the patterns of the sequences, as well as their graphs. Furthermore, their associated Pisano Periods are calculated using mod3 and mod11, showing a correlation between the sequences and the modulus used to calculate the period, albeit, with some unexpected results concerning mod11. Finally, the calculated Pisano Periods are used to circumscribe geometric patterns, showing that each graph for each sequence is unique, but there are some shared characteristics between them.