OEIS A000102: Compositions with bounded parts, binary runs, and Lyndon shadows

We explore A000102, the OEIS sequence counting compositions of n with parts at most four. We’ll unpack why the early terms look the way they do, examine the tidy seven-term recurrence and its generating function, and touch on the indexing convention OEIS uses. A surprising bridge then appears: A000102 also counts binary strings of length n whose longest run of zeros is exactly three. We’ll walk through a compact Python approach that only keeps the last seven values and see how that mirrors the recurrence. The episode then threads in Lucas’s work on binary Lyndon words and hints at deeper connections, including a convolution link to Tribonacci and Tetranacci numbers. Tune in for a crisp voyage from simple rules to rich, interconnected math.

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