Generalized Fibonacci sequences and their properties

Let $F_n(k)$ be the generalized Fibonacci number defined by (with $F_i(k)$ abbreviated to $F_i$): $F_n=F_{n-1}+F_{n-2}+\cdots +F_{n-k}$, for $n \geq k$,andtheinitialvalues$(F_0,F_1,…,F_{k-1})$\mathrm{.}Let$B_n(k,j)$be$F_n(k)$withinitialvaluesgivenby$F_j = 1$and,for$i<j$and$j<i<k$,$F_i = 0$. This paper shows that any $F_n(k)$ can be expressed as the sum of $B_n(k,j)$s. This paper also expresses $B_n(k,j)$ and $F_n(k)$ as finite sums, derives some properties and evaluates their 2-adic order for a range of values of $k,j$ and $n$ and those of $B_n(3,j)$ and $B_n(4,j)$ for most values of $j$ and $n$.