On the Representation of Integers in Nonadjacent Form

This note considers the n-digit representation of the integers of the ring $[ {0,2^n - 2} ]$ in nonadjacent form (NAF), i.e., in radix 2 with coefficients $\{ {1,0, - 1} \}$, such as no two cyclically adjacent coefficients are nonzero. If the interval $[ {0,2^n - 2} ]$ is subdivided into lower, middle, and upper third, it is shown that integers in the lower third are represented as positive, in the upper third as negative, and in the middle third as positive or negative depending upon whether they are even or odd, respectively. Furthermore, there is a one-to-one correspondence between NAF sequences of n digits and integers in $[ {0,2^n - 2} ]$ for n odd, whereas for n even the integers $( 2^n - 1 )/3$ and $2( 2^n - 1 )/3$ are each representable by exactly two NAF sequences.