A Pattern Sequence Approach to Stern's Sequence

Let w be a binary string and let a_w (n) be the number of occurrences of the word w in the binary expansion of n. As usual we let s(n) denote the Stern sequence; that is, s(0)=0, s(1)=1, and for n >= 1, s(2n)=s(n) and s(2n+1)=s(n)+s(n+1). In this note, we show that s(n) = a_1 (n) + \sum_{w in 1 (0+1)*} s([w bar]) a_{w1} (n) where w bar denotes the complement of w (obtained by sending 0 to 1 and 1 to 0, and [w] denotes the integer specified by the word w interpreted in base 2.