A pattern sequence approach to Stern's sequence

Let w ∈ 1{0, 1} * and let a w (n) be the number of occurrences of the word w in the binary expansion of n. Let {s(n)} n0 denote the Stern sequence, defined by 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) + w∈1{0,1} * s([w] 2)a w1 (n) where w denotes the complement of w (obtained by sending 0 → 1 and 1 → 0) and [w] 2 denotes the integer specified by the word w ∈ {0, 1} * interpreted in base 2.