Generalized a:k:m-Fibonacci Numbers
Lovemore Mamombe *
Independent Researcher, Harare, Zimbabwe.
*Author to whom correspondence should be addressed.
Abstract
The a:k:m-Fibonacci sequence is defined recursively by 
. We introduce the generalized a:k:m-Fibonacci sequence 
with arbitrary integers 
and 
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
Keywords: a:k:m-Fibonacci numbers, a:k:m-Lucas numbers, Jacobsthal numbers, k-Fibonacci numbers, k-Lucas numbers, k-Jacobsthal numbers, k-Pell numbers, Lehmer type sequences