Largest values for the Stern sequence

Lansing, Jennifer

Journal of integer sequences, Tome 17 (2014) no. 7

Résumé

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 $w(n) = s(3n)/2$.

Détail

Classification : 11B65
Keywords: stern sequence, largest value