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Patterns in Variations of the Fibonacci Sequence

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Zeckendorf explored the decomposition of integers into sums of nonconsecutive Fibonacci numbers. Thinking of each Fibonacci number in the sequence as being inside a ‘bin’, Zeckendorf’s rule for “legal” decompositions can be reworded as sums of numbers in bins separated by 1 or more bins. Zeckendorf never explored sequences that result when the bins are enlarged to include more than one number or when “legal” decompositions are redefined to require more bins between summands. We do both these things. Another aspect we look at is how these sequences are created. The Fibonacci Sequence has the well known recurrence relation fn = fn−1 +fn−2. Our sequences have a similar single recurrence relation. Also, we have found that given different initial conditions, very similar sequences are produced. We examine where the same terms appear and explain why this is the case. Additionally, we discuss different types of ratios between terms in a given sequence. More specifically, we will examine the quotients of consecutive numbers as well as ratios of summands. Our results show that ratios made up of terms coming from the same relative positions are approximately equal. This research is of particular interest considering that the quotients of terms in the Fibonacci Sequence approach the Golden Ratio, a very special number found in many diverse areas of mathematics. While our quotient values are different, the fact that specific numbers are approached is an important similarity between our sequences and the Fibonacci Sequence. This is not surprising considering that sequences produced in literature have been shown to share significant connections with the Fibonacci Sequence. Nevertheless, the specific question of ratios in any variation of the Fibonacci Sequence has never been explored.

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  • 06/09/2020
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