The characteristic polynomial equals
$ x^{3}−6x^{2}+12x−8$
=$ (x−2)^{3}$ .
$an = γ12^{ n} + γ2n2^{ n} + γ3n^{2}2^{ n}$ for some constants$ γ1, γ2, γ3. $
By substituting n = 0, 1, 2 into this equation, we obtain the linear system
$γ1 = −5$ ,
$2γ1 + 2γ2 + 2γ3 = 4$
$4γ1 + 8γ2 + 16γ3 = 88.$
The solution is these equations $γ1 = −5 , c2 = 1/2, c3 = 13/2,$
and therefore $an = −5 · 2^{ n} + n · 2^{ n−1} + 13 · n^{ 2} · 2^{ n−1}$ for n = 0, 1, 2, 3, . . . .
