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Math Help - Fibonacci Sequence Formula

  1. #1
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    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?
    Last edited by mikewienerm; April 1st 2013 at 04:35 PM.
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  2. #2
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    Re: Finding Formula for given sequence

    Quote Originally Posted by mikewienerm View Post
    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
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  3. #3
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    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.
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  4. #4
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    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
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  5. #5
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    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.
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  6. #6
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    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
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