CONVERGENCES OF THE BI-PERIODIC HORADAM SEQUENCES AND THE BI-PERIODIC LUCAS–HORADAM MATRIX

CONVERGENCES OF THE BI-PERIODIC HORADAM SEQUENCES AND THE BI-PERIODIC LUCAS–HORADAM MATRIX

This paper explores extensions to the Fibonacci and Lucas sequences, examining modifications to the recursive equation and alterations to initial values. Building on previous work detailed in [J. P. Ascaño and E. N. Gueco, The Bi-periodic Fibonacci-Horadam Matrix, Integers 21 (2021), #A29], our focus centers on generalizations of this sequence, particularly the bi-periodic Horadam sequences. These sequences are characterized by initial values corresponding to bi-periodic Fibonacci and bi-periodic Lucas sequences. The investigation extends to studying limits of term ratios, thereby generalizing the concept of limits and the golden ratio seen in classical versions. To enhance comprehension, we introduce three new matrices, with one being the bi-periodic Lucas-Horadam matrix denoted as H. Subsequently, this matrix is utilized to derive various properties of the bi-periodic Horadam sequences, including the exploration of summation identities.