File:K m Fib.pdf

For the integers ${\displaystyle k=1,2,\ldots ,}$ ${\displaystyle m=0,1,2,\ldots ,}$ we define the k-step sum and m-step gap Fibonacci sequence, whose n -th term is given by the recurrence relation ${\displaystyle f_{n}=f_{n-m-1}+f_{n-m-2}+\cdots +f_{n-m-k}=\sum _{j=1}^{k}{f_{n-m-j}},}$ for every ${\displaystyle n\geq k+m+1}$, with ${\displaystyle f_{1}=f_{2}=\ldots =f_{k+m}=1.}$

This recurrence formula generates the 𝑛-th term of the sequence as the sum of 𝑘 successive previous terms starting the sum at the 𝑚-th previous term.

Reference : Adam M., Assimakis N., “k-step sum and m-step gap Fibonacci sequence”, ISRN Discrete Mathematics, vol. 2014, Article ID 374902, 7 pages, http://dx.doi.org/10.1155/2014/374902, 2014.

Demo Link : http://elnsite.teilam.gr/kmfib/