dc.contributor.author Persinger, Carl Allan en_US dc.date.accessioned 2016-02-01T18:05:52Z dc.date.available 2016-02-01T18:05:52Z dc.date.issued 1962 en_US dc.identifier.uri http://hdl.handle.net/10919/64732 dc.description.abstract Early in the thirteenth century, Leonardo de Pisa, or, Fibonacci, introduced his famous rabbit problem, which may be stated simply as follows: assume that rabbits reproduce at a rate such that one pair is born each month from each pair of adults not less than two months old. If one pair is present initially, and if none die, how many pairs will be present after one year? The solution to the problem gives rise to a sequence {Un} known as the Classical Fibonacci Sequence. {Un} is defined by the recurrence relation Un = Un-1 + Un-2, n ≥ 2, U₀ = 0, U₁ = 1 Many properties of this sequence have been derived. A generalized sequence {Fn} can be obtained by retaining the law of recurrence and redefining the first two terms as F₁ = p', F₂ = p' + q' for arbitrary real numbers p' and q'. Moreover, by defining H₁ = p+iq, H₂ = r+is, p,q,r and s real, a complex sequence is determined. Hence, all the properties of the classical sequence can be extended to the complex case. By reducing the classical sequence by a modulus m, many properties of the repeating sequence that results can be derived. The Fibonacci sequence and associated golden ratio occur in communication theory, chemistry, and in nature. en dc.format.extent 58 leaves en_US dc.format.mimetype application/pdf en_US dc.language.iso en_US en_US dc.publisher Virginia Polytechnic Institute en_US dc.rights This Item is protected by copyright and/or related rights. Some uses of this Item may be deemed fair and permitted by law even without permission from the rights holder(s), or the rights holder(s) may have licensed the work for use under certain conditions. For other uses you need to obtain permission from the rights holder(s). en_US dc.subject.lcc LD5655.V855 1962.P477 en_US dc.subject.lcsh Fibonacci numbers en_US dc.title Fibonacci sequences en_US dc.type Thesis en_US dc.contributor.department Mathematics en_US dc.description.degree Master of Science en_US dc.identifier.oclc 22537570 en_US thesis.degree.name Master of Science en_US thesis.degree.level masters en_US thesis.degree.grantor Virginia Polytechnic Institute en_US thesis.degree.discipline Mathematics en_US dc.type.dcmitype Text en_US
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