Recurrence sequences in the hyperbolic Pascal triangle corresponding to the regular mosaic $\{4,5\}

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 ($\alpha\in\mathbb{N}^+$).