Hadamard transforms of three-term sequences

) ( ) ( 2 q q x x x Tr x f + + = and A = {ai} where ) ( i i f a α = . In reference [1], the sequence A is conjectured to have a two-level autocorrelation for all n > 1 and this conjecture has been verified for 23 3 ≤ ≤ n . In this note, we present an interesting formula for the Hadamard transform of the sequence A or, equivalently, for the function f(x). The Hadamard transform of f(x) is defined by