k-Step Sum and m-Step Gap Fibonacci Sequence

Abstract

For two given integers k, m, we introduce the k-step sumand m-step gap Fibonacci sequence by presenting a recurrence formula that generates the nth term as the sum of k successive previous terms starting the sum at the mth previous term. Known sequences, like Fibonacci, tribonacci, tetranacci, and Padovan sequences, are derived for specific values of k, m. Two limiting properties concerning the terms of the sequence are presented. The limits are related to the spectral radius of the associated {0,1}-matrix.