The Binet formula, sums and representations of generalized Fibonacci \(p\)-numbers (Q2426449)

The generalized Fibonacci sequence \({F_p(n)}\) are defined by the following equation for \(n>p+1\) \[ F_p(n)=F_p(n-1)+F_p(n-p-1), \] with initial conditions \[ F_p(1)=F_p(2)=\cdots=F_p(p)=F_p(p+1)=1. \] For \(p=1\), this is the usual Fibonacci sequence. The author investigates the generalized Binet formula, sums, combinatorial representations and generating functions of the generalized Fibonacci sequence \({F_p(n)}\) by using the matrix method.