##
**
Largest Values for the Stern Sequence
**

###
Jennifer Lansing

Department of Mathematics

University of Illinois at Urbana Champaign

1409 W. Green St

Urbana, IL 61801

USA

**Abstract:**

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*(3*n*) is always even.
We introduce and give some properties of the related sequence defined
by *w*(*n*) = *s*(3*n*)/2.

**
Full version: pdf,
dvi,
ps,
latex
**

(Concerned with sequences
A002487
A240388.)

Received April 11 2014;
revised version received June 17 2014.
Published in *Journal of Integer Sequences*, July 1 2014.

Return to
**Journal of Integer Sequences home page**