OEIS A000226: Triangles, Trees, and the Web of Unicyclic Graphs

We explore A000226, the count of unlabeled connected n-node graphs with exactly one 3-cycle (a triangle). We reveal its surprising equivalences with rooted trees on n+1 nodes where the root has degree 3, forests of three rooted trees, and unicyclic graphs with a triangle. We'll trace its history from Riordan through Sloan and Plouffe, examine the generating function tied to rooted trees (A000081), and discuss asymptotics and computational approaches, with pointers to the rich OEIS network of related sequences.

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