Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.5

Largest Values for the Stern Sequence

Jennifer Lansing
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.

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(Concerned with sequences A002487 A240388.)

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

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