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Golden Sequence
Alternate names
Notation
F_n
Sequence description
sequence in which each term is the sum of the two previous terms with F_0 = 0, F_1 = 1, F_n = F_(n - 1) + F_(n - 2)
Sequence terms
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ...
Formula
a_n = (ϕ^n - (-ϕ)^(-n))/sqrt(5)
Recurrence relation
a_0 = 0
a_1 = 1
a_n = a_(n - 2) + a_(n - 1)
Ordinary generating function
sum_(n=0)^∞a_nx^n = x/(-x^2 - x + 1)
Exponential generating function
sum_(n=0)^∞(a_nx^n)/(n!) = (2 e^(x/2) sinh((sqrt(5) x)/2))/sqrt(5)
Dirichlet generating function
sum_(n = 1)^∞a_n/n^s = (Li_s(ϕ) - Li_s(-1/ϕ))/sqrt(5)
Program
a_n = Fibonacci[n]
Table program
Fibonacci[Range[0, n]]
Plot