The Stern sequence (also called Stern’s diatomic sequence), is defined by the recurrence relations s(0) = 0, s(1) = 1, and in general by s(2n) = s(n), and s(2n + 1) = s(n) + s(n + 1). In this note we prove a new identity for Stern’s sequence. In particular, we show that if e and a are nonnegative integers, then for any integer r with 0 ≤ r ≤ 2e, we have s(r)s(2a + 5) + s(2e ? r)s(2a + 3) = s(2e(a + 2) + r) + s(2e(a + 1) + r). It seems that this is the first correlation–type identity concerning Stern’s sequence in the literature.
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