A binary de Bruijn sequence of order n can be generated by an n-stage nonlinear feedback shift register with the feedback function of the form \(x_0\oplus g(x_1,x_2,\ldots ,x_{n-1})\) . In this paper, based on the generating function theory for the weights of de Bruijn sequences proposed by Coppersmith et al., it is shown that the expectation and the variance of the weight of g are \(2^{n-2}+\frac{1}{2} e_{n,1}\) and \(2^{n-3}-\frac{1}{4} e_{n,1}-\frac{1}{2} e_{n,2}\) , respectively, where \(e_{n,1}\) and \(e_{n,2}\) denote the number of cycles of the n-stage pure circulating register with weight 1 and 2, respectively. The expectation and the variance are close to \(2^{n-2}\) and \(2^{n-3}\) when n is large enough, respectively, which shows that the weight distribution of g is very dispersed.
