OEIS A000100: The simple sequence with a surprisingly rich web of connections

We explore A000100, which counts compositions of n with maximum part 3 and the binary strings of length n−1 whose longest zero run is exactly 2. We uncover links to Fibonacci and Tribonacci numbers, see how convolution ties the sequences together, relate to k-step Fibonacci numbers, and learn about the generating function that encodes the whole story. This episode shows how a single, deceptively simple sequence reveals a tapestry of ideas across combinatorics and number theory.

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