Let ϕ = (ϕi)i ≥ 1 and ψ = (ψi)i ≥ 1 be two arbitrary sequences with ϕ1=ψ1. Let
Aϕ,ψ(n) denote the matrix of order n with entries
ai,j, 1 ≤ i, j ≤ n by setting a1,j=ϕj and
ai,1=ψi for 1 ≤ i ≤ n, and where
ai,j=ai−1,j−1+ai−1,j, for 2 ≤ i, j ≤ n. It is of
interest to evaluate the determinant of Aϕ,ψ(n), 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:
| α(k)=(1, 1, …, 1k − times,0, 0, 0, …) , |
| χ(k)=(1k, 2k, 3k, …, ik, …), |
| ξ(k)=(1, k, k2, …, ki−1, …), (a geometric sequence) |
| γ(k)=(1, 1+k, 1+2k, …, 1+(i−1)k, …). (an arithmetic sequence) |
For some sequences of the above type the inverse of Aϕ,ψ(n) is found. In the final part
of this paper, the determinant of a generalized Pascal triangle
associated to Fibonacci sequence is found.
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