LEMMA ON THE DETERMINANTAL RELATION OF SEQUENCES
DOI:
https://doi.org/10.31471/2304-7399-2025-20(76)-82-93Keywords:
recurrence relations, matrices, determinants, sequencesAbstract
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.
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