Journal of the
Korean Mathematical Society JKMS

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 $ D\left[\begin{smallmatrix} 0\\ \mathbf{x} \end{smallmatrix}\right]$ is perpendicular to every truncated Fibonacci (truncated Lucas resp.) sequence of the second kind where $$D={\rm diag}((-1)^0 , (-1)^1 , (-1)^2 , \ldots).$$

Keywords: $(\pm 1)$-invariant sequence, truncated Fibonacci sequence of the second kind

MSC numbers: Primary 11B39, 11B50