Largest Values for the Stern Sequence
Department of Mathematics
University of Illinois at Urbana Champaign
1409 W. Green St
Urbana, IL 61801
In 1858, Stern introduced an array, later called the diatomic array.
The array is formed by taking two values a and b for the first row,
and each succeeding row is formed from the previous by inserting c+d
between two consecutive terms with values c, d. This array has
many interesting properties, such as the largest value in a row of the
diatomic array is the (r+2)-th Fibonacci number, occurring roughly
one-third and two-thirds of the way through the row. In this paper, we
show each of the second and third largest values in a row of the
diatomic array satisfy a Fibonacci recurrence and can be written as a
linear combination of Fibonacci numbers. The array can be written in
terms of a recursive sequence, denoted s(n) and called the Stern
sequence. The diatomic array also has the property that every third
term is even. In function notation, we have s(3n) is always even.
We introduce and give some properties of the related sequence defined
by w(n) = s(3n)/2.
Full version: pdf,
(Concerned with sequences
Received April 11 2014;
revised version received June 17 2014.
Published in Journal of Integer Sequences, July 1 2014.
Journal of Integer Sequences home page