OEIS A000253: Binary strings containing the pattern 010

We explore A000253, which counts binary strings of length n+2 that contain the substring 010. With offset 0, the sequence starts 0, 1, 4, 11, 27, 63, ... and satisfies a_n = a_{n-1} + a_{n-2} + a_{n-3} + 2^{n-1} for n ≥ 3, with initial a0=0, a1=1, a2=4. The combinatorial heart is a Holger Petersen-style count: split strings into those that already contain 010 and those that first acquire it when extended, yielding the recurrence. We discuss how this concrete counting problem connects to generating functions, linear recurrences (and even higher-order homogeneous forms), and how computers (memoized recursion, matrix methods) help generate large terms. Finally, we touch on cross-references and why these interconnected OEIS entries matter for deeper patterns in combinatorics.

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