Certain matrices related to Fibonacci sequence having recursive entries

Abstract

Let φ = (φ i ) i≥1 and ψ = (ψ i ) i≥1 be two arbitrary sequences with φ 1 = ψ 1 . Let A φ,ψ (n) denote the matrix of order n with entries a i,j , 1 ≤ i, j ≤ n, where a 1,j = φ j and a i,1 = ψ i for 1 ≤ i ≤ n, and where, where one of the sequences φ or ψ is the Fibonacci sequence (i.e., 1, 1, 2, 3, 5, 8, . . .) and the other is one of the following sequences:1, 1, . . . , 1, 0, 0, 0, . . .) ,For some sequences of the above type the inverse of A φ,ψ (n) is found. In the final part of this paper, the determinant of a generalized Pascal triangle associated to the Fibonacci sequence is found.