sum_(n=0)^∞ L_n t^n = (t + 2 t^2)/(1 - t - t^2)

L_n is a sequence with integer values for integer n.

n | L_n 0 | 2 1 | 1 2 | 3 3 | 4 4 | 7 5 | 11 6 | 18 7 | 29 8 | 47 9 | 76 10 | 123

(1/2 (1 + sqrt(5)))^n + (2/(1 + sqrt(5)))^n cos(π n)

2 + n^2 (log^2(ϕ) - π^2/2) + 1/2 π^2 n^3 log(ϕ) + 1/24 n^4 (2 log^4(ϕ) - 6 π^2 log^2(ϕ) + π^4) + O(n^5) (Taylor series)

d/dn(L_n) = ϕ^n log(ϕ) - π ϕ^(-n) sin(π n) - ϕ^(-n) cos(π n) log(ϕ)

L_n = F_(-1 + n) + F_(1 + n)

L_n = (F_(m + n) + (-1)^n F_(m - n))/F_m for (m element Z and m!=0 and n element Z)

L_n = (cos(n π) (-1 + sqrt(5))^n + (1 + sqrt(5))^n)/2^n

L_n = 2^(1 - n) sum_(k=0)^floor(n/2) 5^k binomial(n, 2 k) for (n element Z and n>=0)

L_n = 1/2 sum_(k=0)^∞ (n^k ((i π - csch^(-1)(2))^k + 2 csch^(-1)(2)^k + (-1)^k (i π + csch^(-1)(2))^k))/(k!)

L_n = sum_(k=0)^∞ ((ϕ^(ν_0) csch^(-1)(2)^k + 1/2 ϕ^(-ν_0) (e^(i π ν_0) (i π - csch^(-1)(2))^k + (-1)^k e^(-i π ν_0) (i π + csch^(-1)(2))^k)) (n - ν_0)^k)/(k!)

L_(2 n + 1) = 1/4 (3/2)^(n - 1) integral_0^π (1 + 1/3 sqrt(5) cos(t))^(n - 1) (3 + 5 n + sqrt(5) (1 + n) cos(t)) sin(t) dt for n element Z