It is a consequence of the form of the characteristic equation: if the double root is r, then expanding out x^2 - 2r + r^2 and matching terms we see that a = 2r

It is a consequence of the form of the characteristic equation: if the double root is r, then expanding out x^2 - 2r + r^2 and matching terms we see that a = 2r and b = -r^2; in other words, the recurrence is a(n) = 2r a(n-1) - r^2 a(n-2). Divide by r^n to get the equivalent c(n) = 2c(n-1) - c(n-2), where c(n) = a(n)/r^n But this is the constant difference recurrence: c(n) - c(n-1) = c(n-1) - c(n-2) So c(n) is an arithmetic progression c(n) = x*n + y for some x and y given by the initial conditions. Thus the original sequence is a(n) = c(n) r^n = (x*n + y) r^n.