The determinants of matrices constructed by subdiagonal, main diagonal and superdiagonal

Abstract

The purpose of this article is to prove several evaluations of determinants of matrices, the entries of which are given by the recurrences

$$ a_{i,j} = \left\{ {\begin{array}{*{20}c} {a_{i,j - 2} + a_{i + 1,j - 1} + a_{i + 2,j,} if j \geqslant i + 2;} \\ {a_{i - 2,j} + a_{i - 1,j + 1} + a_{i,j + 2,} if i \geqslant j + 2;} \\ \end{array} } \right. $$

with various choices for main diagonal a _i,i , superdiagonal a _i,i+1 and subdiagonal a _i+1,i.