Rhaly terraced sequences their generalizations, properties and applications

Abstract This paper links terraced matrices with other well-known integer sequences, such as the Hankel matrices and related Fibonacci and Lucas matrices. These, in turn, are connected with related results of Macmahon and Sloane as well as we introduce the r-Terraced matrix as a generalization of the Terraced matrix, along with its symmetric counterpart, the symmetric r-Terraced matrix. We derive key properties of these matrices, including their spectral and Euclidean norms, upper bounds for their spreads, and characteristic polynomials. To validate and exemplify the theoretical findings, we apply them to Fibonacci numbers, providing illustrative examples that strengthen the theory and confirm its accuracy. In addition to the theoretical results, we investigated how the choice of the parameter r and the matrix dimension affect the upper bounds of the spread. Our findings reveal that selecting values of $$r&lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and using lower-dimensional matrices lead to tighter upper bounds while reducing computational complexity. These results highlight the practical benefits of our approach, particularly in optimization-related applications where efficiency is crucial.