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An exploration of nested recurrences using experimental mathematics
Descriptive
TitleInfo
Title
An exploration of nested recurrences using experimental mathematics
Name
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Fox
NamePart
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Nathan Harel
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Nathan Harel Fox
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author
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Zeilberger
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Doron
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Doron Zeilberger
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Advisory Committee
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chair
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Kopparty
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Swastik
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Swastik Kopparty
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Advisory Committee
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internal member
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Komlos
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Janos
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Janos Komlos
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Advisory Committee
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internal member
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Sloane
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Neil
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Neil Sloane
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Advisory Committee
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outside member
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Rutgers University
Role
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degree grantor
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Graduate School - New Brunswick
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Text
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theses
OriginInfo
DateCreated
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2017
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2017-05
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2017
Place
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xx
Language
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eng
Abstract
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Broadly speaking, Experimental Mathematics is the philosophy that computers are a valuable tool that should be used extensively in mathematical research. Here, we apply this philosophy to the study of integer sequences arising from nested recurrence relations. The most widely studied nested recurrence is the Hofstadter Q-recurrence: Q(n) = Q(n − Q(n − 1)) + Q(n − Q(n − 2)). Hofstadter considered this recurrence with the initial condition Q(1) = Q(2) = 1, and the resulting sequence has much apparent structure. But, almost nothing has been rigorously proved about it. Others have modified the recurrence, the initial conditions, or both, to obtain related but more predictable sequences. We follow that vein and prove a number of unrelated theorems about sequences resulting from nested recurrences. Our first results relate to automatically finding (with proof) solutions to nested recurrences that are interleavings of linear-recurrent sequences. We then present a new nested-recurrent sequence whose terms increase monotonically with successive differences zero or one. Finally, we embark on an exploration of strange but predictable behaviors that result when recurrences are given various types of initial conditions.
Subject
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Topic
Mathematics
Subject
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Topic
Sequences (Mathematics)
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Rutgers University Electronic Theses and Dissertations
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ETD_7946
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electronic resource
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application/pdf
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text/xml
Extent
1 online resource (xi, 227 p. : ill.)
Note
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Ph.D.
Note
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Includes bibliographical references
Note
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by Nathan Harel Fox
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Title
Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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NjNbRU
Identifier
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doi:10.7282/T3WD43FW
Genre
(authority = ExL-Esploro)
ETD doctoral
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Rights
RightsDeclaration
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The author owns the copyright to this work.
RightsHolder
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Name
FamilyName
Fox
GivenName
Nathan
MiddleName
Harel
Role
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Type
Permission or license
DateTime
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2017-04-06 15:12:19
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Name
Nathan Fox
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Affiliation
Rutgers University. Graduate School - New Brunswick
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Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
Copyright
Status
Copyright protected
Availability
Status
Open
Reason
Permission or license
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Technical
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2017-04-06T00:19:18
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2017-04-06T00:19:18
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