Largest Values for the Stern Sequence

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. 1 Historical background The Stern sequence, denoted by s(n), satisfies the recurrences s(2n) = s(n) and s(2n+ 1) = s(n+ 1) + s(n) for n ≥ 1, with s(0) = 0 and s(1) = 1. The first few values of this sequence are 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, . . .