OEIS A000247: 2^n - n - 2

In this Deep Dive we explore OEIS A000247, the sequence a(n) = 2^n − n − 2 for n ≥ 2, which starts 0, 3, 10, 25, 56, 119, 246, 501. We’ll unpack the meaning of the closed form, the offset, and how such a simple formula appears in a surprisingly wide range of problems: counting ways to split n+1 labeled balls into two indistinguishable boxes with at least two in each; permutation patterns that avoid 13-2 and contain 23-1 exactly twice; costs of ternary maximum-height Huffman trees; and the special Dick paths with a certain last long ascent. Along the way we’ll see how the OEIS gathers these interpretations, formulas, and connections, revealing the unity behind discrete math and computer science.

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