LEMMA ON THE DETERMINANTAL RELATION OF SEQUENCES - Zatorsky, R. A. (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2025)
In The On-Line Encyclopedia of Integer Sequences (OEIS), relationships between numerical sequences are frequently cited, although these connections mostly arise spontaneously. Often, the source of such relationships lies in second-order, and less frequently third-order, linear recurrence sequences. This article considers numerical sequences associated with infinite linear recurrence relations and proves a lemma on their determinantal relation. To this end, matrices are introduced that generalize the Hessenberg–Toeplitz matrices. These matrices are associated with unordered partitions of a natural number into natural summands. This general, matrix-based approach to the study of sequences naturally establishes a bijection between mutually conjugate partitions of a natural number into natural summands and provides a framework for identifying connections between sequences associated with linear recurrence relations. The article also presents important corollaries of the lemma and illustrates them by examining the connection between the $n-$th term of the Pell number sequence and the Fibonacci sequence, as well as their principal determinants.