On the Stern sequence and a related sequence

In this dissertation, we discuss properties of the Stern sequence, denoted by s(n), and define a related sequence. First, we give a brief historical background and known results. We then discuss the second and third largest values for the Stern sequence, as well as the asymptotics when a value m will first appear in a row in the diatomic array. We also investigate the distribution of values for the Stern sequence, as well as the gaps of the ordered values from a row. After this, we investigate the properties of the related sequence called w(n):=s(3n)/2. We give recurrences for the sequence and find generalized recurrences and a reduction formula. We attempt to find a combinatorial interpretation for w(n), as well as a generating function for the sequence. We also find the largest value of w(n) for a row of its triangular array. We consider sums of w(n) and the average order of magnitude, which is the same average order of the Stern sequence. We also examine the greatest common divisor of consecutive terms, as well as the sequence w(n) modulo 2. Finally, we define a polynomial analogue and investigate some of its properties.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Lansing, Jennifer

Contributors dc:contributor

• Reznick, Bruce A.
• Berndt, Bruce C.
• Zaharescu, Alexandru
• Stolarsky, Kenneth B.

• Stern sequence

Statement dc:rights

• Copyright 2014 Jennifer Lansing

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/49483

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/49483