Abstract
In this paper, we will see that some k -Fibonacci sequences are related to the classical Fibonacci sequence of such way that we can express the terms of a k -Fibonacci sequence in function of some terms of the classical Fibonacci sequence. And the formulas will apply to any sequence of a certain set of k' -Fibonacci sequences. Thus we find k -Fibonacci sequences relating to other k -Fibonacci sequences when σ'k is linearly dependent of .
Highlights
{ } k -Fibonacci sequence Fk,n n≥0 was found by studying the recursive application of two geometrical transformations used in the well-known four-triangle longest-edge (4TLE) partition
There are infinite k -Fibonacci sequences related to an initial k -Fibonacci sequence for a fixed value of “ k ”
3) Second sequence related to the k –Fibonacci sequence:= k′ k k 2 + 3 4) Two first values of “ b ” are b1= 1= Fk,1 and b2 = k 2 +1= Fk,3 5) Two first values of “ a ” are a0= 0= Fk,0 and a1= k= Fk,2 ( ) 6) Recurrence law for the sequences {an},{bn}, and {kn} : pn+1 =k 2 + 2 pn − pn−1
Summary
{ } k -Fibonacci sequence Fk,n n≥0 was found by studying the recursive application of two geometrical transformations used in the well-known four-triangle longest-edge (4TLE) partition. This sequence generalizes the classical Fibonacci sequence [1] [2]
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