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    Lucas Sequence

    Notation

    L_n

    Sequence description

    sequence in which each term is the sum of the two previous terms with L_1 = 1, L_2 = 3, L_n = L_(n - 1) + L_(n - 2)

    Sequence terms

    1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, ...

    Formula

    a_n = (-ϕ)^(-n) + ϕ^n

    Recurrence relation

    a_1 = 1

    a_2 = 3

    a_n = a_(n - 2) + a_(n - 1)

    Ordinary generating function

    sum_(n=0)^∞a_nx^n = (x (2 x + 1))/(-x^2 - x + 1)

    Exponential generating function

    sum_(n=0)^∞(a_nx^n)/(n!) = e^(-x/ϕ) + e^(x ϕ)

    Dirichlet generating function

    sum_(n = 1)^∞a_n/n^s = Li_s(-1/ϕ) + Li_s(ϕ)

    Table program

    LinearRecurrence[{1, 1}, {1, 3}, n]

    Plot

    Plot

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