(1) をやってみた

αβ = -1
S1 = α+ β = 4
S2 = α^2 + β^2 = (α+β)^2 - 2αβ = 16 - 2 * (-1) = 18
S3 = α^3 + β^3 = (α+β)(α^2+β^2-αβ) = 4 * (18 - (-1)) = 76
Sn = α^n + β^n = (4α+1)α^(n-2) + (4β+1)β^(n-2) = 4S_{n-1} + S_{n-2}