$(\pm 1)$-Invariant Sequences and Truncated Fibonacci Sequences of the Second Kind

Abstract

In this paper we present another characterization of $(\pm 1)$-invariant sequences. We also introduce truncated Fibonacci and Lucas sequences of the second kind and show that a sequence $\mathbf{x} \in \mathbf{R}^\infty$ is $(-1)$-invariant($1$-invariant resp.) if and only if \begin{math} D\bigl[\begin{smallmatrix} 0\\ \mathbf{x} \end{smallmatrix}\bigr]\end{math} is perpendicular to every truncated Fibonacci(truncated Lucas resp.) sequence of the second kind where \begin{center} $D=\textup{diag}((-1)^0 , (-1)^1 , (-1)^2 , \ldots)