{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/64732"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/64732","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Fibonacci sequences","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 {U<sub>n</sub>} known as the Classical Fibonacci Sequence. {U<sub>n</sub>} is defined by the recurrence relation U<sub>n</sub> = U<sub>n-1</sub> + U<sub>n-2</sub>, n ≥ 2, U₀ = 0, U₁ = 1 Many properties of this sequence have been derived. A generalized sequence {F<sub>n</sub>} 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.","abstract_html":"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 {U&lt;sub&gt;n&lt;/sub&gt;} known as the Classical Fibonacci Sequence. {U&lt;sub&gt;n&lt;/sub&gt;} is defined by the recurrence relation U&lt;sub&gt;n&lt;/sub&gt; = U&lt;sub&gt;n-1&lt;/sub&gt; + U&lt;sub&gt;n-2&lt;/sub&gt;, n ≥ 2, U₀ = 0, U₁ = 1 Many properties of this sequence have been derived. A generalized sequence {F&lt;sub&gt;n&lt;/sub&gt;} can be obtained by retaining the law of recurrence and redefining the first two terms as F₁ = p&#x27;, F₂ = p&#x27; + q&#x27; for arbitrary real numbers p&#x27; and q&#x27;. 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.","abstract_has_math":false,"creators":["Persinger, Carl Allan"],"institution":"Virginia Polytechnic Institute","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1962,"date_issued":"1962","date_published":"1962","updated_at":"2026-07-22T22:19:33Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/64732","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Persinger, Carl Allan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2016-02-01T18:05:52Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2016-02-01T18:05:52Z"]},{"key":"dc:date.issued","label":"Date","values":["1962"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/64732"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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 {U<sub>n</sub>} known as the Classical Fibonacci Sequence. {U<sub>n</sub>} is defined by the recurrence relation U<sub>n</sub> = U<sub>n-1</sub> + U<sub>n-2</sub>, n ≥ 2, U₀ = 0, U₁ = 1 Many properties of this sequence have been derived. A generalized sequence {F<sub>n</sub>} 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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Fibonacci sequences"]}]}],"canonical_facts":{"dc:contributor.department":["Mathematics"],"dc:creator":["Persinger, Carl Allan"],"dc:date.accessioned":["2016-02-01T18:05:52Z"],"dc:date.available":["2016-02-01T18:05:52Z"],"dc:date.issued":["1962"],"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 {U<sub>n</sub>} known as the Classical Fibonacci Sequence. {U<sub>n</sub>} is defined by the recurrence relation U<sub>n</sub> = U<sub>n-1</sub> + U<sub>n-2</sub>, n ≥ 2, U₀ = 0, U₁ = 1 Many properties of this sequence have been derived. A generalized sequence {F<sub>n</sub>} 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."],"dc:description.degree":["Master of Science"],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/64732"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Fibonacci sequences"],"dc:type":["Thesis"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute"]},"updated_at":"2026-07-22T22:19:33Z"}