OEIS A00078: The Tetranacci Numbers

We explore A00078, the tetranacci sequence defined by T(n) = T(n-1) + T(n-2) + T(n-3) + T(n-4) with initial values T(0)=T(1)=T(2)=0 and T(3)=1. We trace its growth, derive its generating function G(x) = x^3 / (1 - x - x^2 - x^3 - x^4), and uncover how this four-term recurrence connects to counting compositions with parts 1–4, binary strings avoiding 1111, and polygon triangulations. We’ll also discuss closed-form-like expressions via roots of the characteristic polynomial, its place in the family of n-step Fibonacci sequences, and the rich web of OEIS links that illuminate the deeper structure behind a simple rule.

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