No label defined (Q6574108)

The Fibonacci sequence is a well-known example of a second-order recurrence sequence, defined by the recurrence relation \( F_0=0 \), \( F_1=1 \), and \( F_n=F_{n-1}+F_{n-2} \) for all \( n\ge 2 \). The bi-periodic Fibonacci sequence is a generalization of the Fibonacci sequence, defined for any integer \( n\ge 2 \) by the recurrence relation \( q_n=a^{\xi(n-1)}b^{\xi(n)}q_{n-1}+q_{n-2}, \) with initial values \( q_0=0 \) and \( q_1=1 \), where \( a \) and \( b \) are nonzero real numbers, and \( \xi(n)=n-2\lfloor n/2 \rfloor \), that is, \( \xi(n)=0 \) when \( n \) is even and \( \xi(n)=1 \) when \( n \) is odd. In the paper under review, the authors provide a combinatorial interpretation for the bi-periodic Fibonacci numbers squared in terms of weighted linear tilings involving two types of tiles. This interpretation allows the authors to establish combinatorial proofs for various identities concerning bi-periodic Fibonacci numbers squared.