k-Step Sum and m-Step Gap Fibonacci Sequence

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 n th term as the sum of k successive previous terms starting the sum at the m th 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.