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Every binary De~Bruijn sequence of order n satisfies a recursion0=x_n+x_0+g(x_{n-1}, ..., x_1). Given a function f on (n-1) bits, let N(f; r)be the number of functions generating a De Bruijn sequence of order n which areobtained by changing r locations in the truth table of f. We prove a formulafor the generating function \sum_r N(\ell; r) y^r when \ell is a linearfunction.  The proof uses a weighted Matrix Tree Theorem and a description of thein-trees (or rooted trees) in the n-bit De Bruijn graph as perturbations of theHamiltonian paths in the same graph.

Counting De Bruijn sequences as perturbations of linear recursions