OEIS A000201: A Fibonacci-family recurrence and its 0-1 matrix interpretation

We examine A000201, defined by a_n = a_{n-1} + a_{n-2} - 2 with a_0 = 4 and a_1 = 3. The sequence sits in the Fibonacci–Lucas family via a_n = F_{n-1} + F_n + 2 and a_n = L_n + 2. A striking combinatorial meaning, due to Vladimir Shevelov, counts n×n binary matrices with exactly two 1s per row and column whose 1s lie only on the diagonals I, P, and P^{-1} (I is the identity and P is the cycle permutation). The count equals the permanent of I + P + P^{-1}, linking linear algebra, combinatorics, and matrix theory. We also touch ordinary and exponential generating functions and point to key references for further exploration.

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