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Math Help - Closed Form

  1. #1
    Junior Member
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    Closed Form

    How could one write a closed form function of the following..

    F(0) = 0
    F(1) = 1
    F(2) = 4
    F(n) = 3 * (F(n-1) F(n-2)) + F(n-3) + 3

    Computing
    f(3) = 3 * (4-1) + 0 + 3 = 12
    f(4) = 3 * (12-4) + 1 + 3 = 28
    f(5) = 3 * (28-12) + 4 + 3 = 55

    I am not seeing a pattern here any help would be appreciated.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Consider G(x)=F_1x+F_2x^2+.... You have :

    G(x)/x=F_1+F_2x+F_3x^2+...=1+F_2x+F_3x^2+...

    (G(x)/x-1)/x = F_2+F_3x+F_4x^2... = 4+F_3x+F_4x^2...

    ((G(x)/x-1)/x-4)/x = F_3+F_4x + F_5x^2 +...

    Now ((G(x)/x-1)/x-4)/x = \sum_{j=0}^\infty F(j+3)x^j = \sum_{j=0}^\infty (3F(j+2)-3F(j+1)+F(j)+3)x^j

    = 3(G(x)/x-1)/x-3G(x)/x+G(x)+\frac{3}{1-x}.

    Solve for G(x) and express it as partial fractions; the coefficient of x^n will be F_n.
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