golden sequence | Lamé's sequence

F_n

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)

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ...

a_n = (ϕ^n - (-ϕ)^(-n))/sqrt(5)

a_0 = 0

a_1 = 1

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

sum_(n=0)^∞a_nx^n = x/(-x^2 - x + 1)

sum_(n=0)^∞(a_nx^n)/(n!) = (2 e^(x/2) sinh((sqrt(5) x)/2))/sqrt(5)

sum_(n = 1)^∞a_n/n^s = (Li_s(ϕ) - Li_s(-1/ϕ))/sqrt(5)

a_n = Fibonacci[n]

Fibonacci[Range[0, n]]