OEIS A000287: Rooted polyhedral graphs with n edges

We explore A000287, the number of rooted polyhedral graphs with n edges. Rooted means a distinguished edge on the polyhedral skeleton, so counting distinguishes shapes that would be equivalent without the root; the sequence begins with n = 6 → 1, n = 7 → 0, and then jumps to much larger values, with a striking parity pattern (odd exactly when n+2 is a power of two). We sketch the main tools used to study it—explicit recurrences (including Plouffe’s four-step recurrence), generating functions, and differential equations for the generating function—along with the large-n asymptotics that reveal connections to continuous analysis (involving constants like pi) behind this discrete counting problem.

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