Alexander Bass
This note is one of many taken during 2024

Fibonacci Sequence

The Fibonacci sequence is defined as

F1=F2=1Fn+2=Fn+1+Fn \begin{align*} F_{1} =& F_{2} = 1\\ F_{n+2} =& F_{n+1} + F_{n} \end{align*}

Are there any Fibonacci numbers FnF_{n} such that Fn+1>2FnF_{n+1} > 2F_{n}?

Fn+1=Fn+Fn−1 F_{n+1} = F_{n} + F_{n-1}

Substitute into inequality

Fn+Fn−1>2Fn F_{n} + F_{n-1} > 2F_{n} Fn−1>Fn F_{n-1} > F_{n}

Which is never true for the sequence.

Another question: which Fibonacci numbers FnF_{n} are such that Fn+2>2FnF_{n+2} > 2F_{n}?

Again substitute

Fn+1+Fn>2Fn F_{n+1} + F_{n} > 2F_{n} Fn+1>Fn F_{n+1} > F_{n}

Which is true for all Fibonacci numbers except for when n=1n=1.

In a similar spirit, it is obvious that

Fn=Fn−1+Fn−2 F_n = F_{n-1} + F_{n-2}

But this form can be reduced further to be a sum of Fn−2F_{n-2} and Fn−3F_{n-3}

Fn=Fn−1+Fn−2=Fn−2+Fn3+Fn−2=2Fn−2+Fn−3and again=2Fn−3+2Fn−4+Fn−3=3Fn−3+2Fn−4 \begin{align*} F_n =& F_{n-1} + F_{n-2}\\ =& F_{n-2} + F_{n_3} + F_{n-2}\\ =& 2F_{n-2} + F_{n-3}\\ & \text{and again}\\ =& 2F_{n-3} + 2F_{n-4} + F_{n-3}\\ =& 3F_{n-3} + 2F_{n-4} \end{align*}

The coefficients on the reduced forms are interesting to me. Given coefficients ak,bka_{k},b_{k} in a specific level of reduced Fibonacci, find ak+1,bk+1a_{k+1},b_{k+1}

akFn−1+bkFn−2=ak(Fn−2+Fn−3)+bkFn−2=(ak+bk)Fn−2+akFn−3 a_k F_{n-1} + b_k F_{n-2} = a_k (F_{n-2} + F_{n-3}) + b_k F_{n-2} = (a_k +b_k)F_{n-2} + a_k F_{n-3}

We define a1=b1=1a_1 =b_1 = 1. and,

ak+1=ak+bkbk+1=ak \begin{align*} a_{k+1} =& a_k + b_k \\ b_{k+1} =& a_k \end{align*} ak+2=ak+1+bk+1=ak+1+ak a_{k+2} = a_{k+1} + b_{k+1} = a_{k+1} + a_{k}

Notice something peculiar about the above identity: it’s also the Fibonacci sequence!

This allows us to define a specific level of reduced Fibonacci symbolically. Let k>1k > 1

Fn=akFn−k+1+bkFn−k F_n = a_k F_{n-k+1} + b_k F_{n-k} Fn=FkFn−k+1+Fk−1Fn−k F_n = F_k F_{n-k+1} + F_{k-1} F_{n-k}

As an example n=10n=10 and k=4k=4

F10=F4F7+F3F6 F_{10} = F_{4}F_{7} + F_{3} F_{6} 55=3⋅13+2⋅8 55 = 3\cdot 13 + 2 \cdot 8
This note is one of many taken during 2024