# Fibonacci Sequence Formula

• Apr 1st 2013, 05:27 PM
mikewienerm
Finding Formula for given sequence
I have the following sequence: 0 2 3 5 8 13 21 34..........
This is very close to the fibonacci sequence, but not exactly. Any ideas?
• Apr 1st 2013, 08:01 PM
Gusbob
Re: Finding Formula for given sequence
Quote:

Originally Posted by mikewienerm
I have the following sequence: 0 2 3 5 8 13 21 34..........
This is very close to the fibonacci sequence, but not exactly. Any ideas?

$F_0 = 0, F_1=2, F_3=3, F_n=F_{n-1}+F_{n-2}\quad\forall n\geq 3$
• Apr 2nd 2013, 06:38 AM
mikewienerm
Re: Finding Formula for given sequence
This is a recursive formula. I was hoping someone could help me find a forumla where I can find F(n) without having to find the previous values. For example, finding F(50) would be very tedious.
• Apr 2nd 2013, 06:46 AM
Gusbob
Re: Fibonacci Sequence Formula
After the first term, this sequence is exactly the same as the Fibonacci sequence (with the first three terms truncated). Surely you can use the same formula to calculate terms $\geq 2$. I.e. if the fibonacci sequence is $F_1,F_2,...$ and your sequence here is $T_1,T_2,...$, set

$T_{n}=F_{n+2}=\frac{\varphi^{n+2}-\psi^{n+2}}{\sqrt5}$

for $n\geq 2$
• Apr 2nd 2013, 07:08 AM
HallsofIvy
Re: Fibonacci Sequence Formula
What Gusbob said, of course!

But also you can apply the same ideas that lead to the "usual" Fibonacci formula. If you "look for" a solution of the form $F_n= r^n$, the equation becomes $r^n= r^{n-1}+ r^{n-2}$. Dividing through by $r^{n-2}$ gives you $r^2= r- 1$ or $r^2- r+ 1= 0$. Solving that $r= \frac{1\pm\sqrt{5}}{2}$. That is, $F_n= A\left(\frac{1+\sqrt{6}}{2}\right)^n+ B\left(\frac{1- \sqrt{6}}{2}\right)^n$ for some constants A and B. That is exactly the solution you would get for the "usual" Fibonnacci formula. Putting $F_1= A\frac{1+ \sqrt{5}}{2}+ B\frac{1- \sqrt{6}}{2}= 0$ and $A\left(\frac{1+ \sqrt{5}}{2}\right)^2+ B\left(\frac{1- \sqrt{5}}{2}\right)^2= 2$ rather than setting them equal to the "usual" 1, 1, solve for A and b.
• Apr 6th 2013, 05:10 AM
isparks
Re: Fibonacci Sequence Formula
Let G represent the Golden Ratio
Then Fn=(G^n+(G-1)^n))/Sqrt 5
Where ^ represents is to the power of
Another method is to use the Binomial Expansion by observing Pascal Triangle