Generalized a:k:m-Fibonacci Numbers

The a:k:m-Fibonacci sequence is defined recursively by ::,::,=::,+::,,,,,≥1::,=1,::,=. We introduce the generalized a:k:m-Fibonacci sequence ::,ℎ::,=ℎ::,+ℎ::,,≥1,with arbitrary integers ℎ::,and ℎ::,, and study some important properties relating to the Pascal type triangle generated from this sequence. The results are extended to negative values of and , an important concept which brings on board Lehmer type sequences. Most importantly, generalized a:k:m-Fibonacci sequences provide a means to unify ideas that are otherwise treated independently. The theory of a:k:m-Fibonacci numbers is therefore a unification theory