  Recently, a new generalization of Pascal's triangle, the so-called hyperbolic
Pascal triangles were introduced. The mathematical background goes back to the
regular mosaics in the hyperbolic plane. In this article, we investigate the
paths in the hyperbolic Pascal triangle corresponding to the regular mosaic
$\{4,5\}$, in which the binary recursive sequences $f_{n}=\alpha f_{n-1}\pm
f_{n-2}$ are represented ($\alpha\in\mathbb{N}^+$).
