Today on The Deep Dive we explore OEIS A000340: the recursively defined sequence with a(0)=1 and a(n)=3·a(n−1)+n+1. We trace its explicit closed form, its generating function, and the other recurrences OEIS lists that define the same numbers. We'll also place A000340 in a broader mathematical context—its connections to sums of powers of 3, appearance in combinatorial triangles, and its classic roots in Sloan's handbooks—along with some modern applications in chip-firing games, permutation patterns, and number expansions. A compact example of how different descriptions illuminate different properties of the same sequence.

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