K-Step Sum and m-Step Gap Fibonacci Sequence

Maria Adam,Nicholas Assimakis

doi:10.1155/2014/374902

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.

Highlights

It is well-known that the Fibonacci sequence, the Lucas sequence, the Padovan sequence, the Perrin sequence, the tribonacci sequence, and the tetranacci sequence are very prominent examples of recursive sequences, which are defined as follows
We introduce k-step sum and m-step gap Fibonacci sequence, where the nth term of the sequence is the sum of the k successive previous terms starting at the mth previous term, using 1’s as initial conditions
A recurrence formula was presented generating the nth term of the sequence as the sum of k successive previous terms starting the sum at the mth previous term

Summary

Introduction

It is well-known that the Fibonacci sequence, the Lucas sequence, the Padovan sequence, the Perrin sequence, the tribonacci sequence, and the tetranacci sequence are very prominent examples of recursive sequences, which are defined as follows. The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, . Are derived by the recurrence relation pn = pn−2 + pn−3, n ≥ 4, with p1 = 3, p2 = 0, and p3 = 2, [2, A001608]. Both Fibonacci and Lucas numbers as well as both Padovan and Perrin numbers satisfy the same recurrence relation with different initial conditions. Further the closed formula of the nth term of the sequence is given and the ratio of two successive terms tends to the spectral radius of the associated {0, 1}-matrix

Definition of k-Step Sum and m-Step Gap Fibonacci Sequence

Limiting Properties of k-Step Sum and mStep Gap Fibonacci Sequence

Conclusions