Apologies for the uninformative title, this is a relatively specific question so it was hard to title.
I'm solving the following recurrence relation:
$a_{n} + a_{n-1} - 6a_{n-2} = 0$
With initial conditions $a_{0} = 3$ and $a_{1} = 1$
And I have it mostly figured out except for the very last part.
My working:
We have characteristic equation $s^2 + s - 6 = 0$ This factorises to $(s+3)(s-2)$
Hence we have roots $s=-3$ and $s=2$
and hence the solution has the form $a_{n} = -x3^n + y2^n$
We sub in the initial conditions:
$a_{0} = x + y = 3$
$a_{1} = -3x+2y = 1$
And solving this system we have solutions:
$x = 1$ and $y = 2$
Hence subbing this back to what we work out to be the general form of the solution:
$a_{n} = (-1)3^n + (2)2^n$
$a_{n} = (-3)^n + (4)^n$ Correct?
But it is incorrect, the correct solution is:
$a_{n} = (-3)^n + 2^{n+1}$
I don't understand where the $2^{n+1}$ came from. What am I missing here?