Easy recursion problem:
Let a, b, c be positive integers. Define a sequence as follows: $x_0=a, x_1=b, x_{n+1}=cx_nx_{n-1}$. Find a nice formula for $x_n$.

Question

Easy recursion problem:
Let a, b, c be positive integers. Define a sequence as follows: $x_0=a, x_1=b, x_{n+1}=cx_nx_{n-1}$. Find a nice formula for $x_n$.

kinggeorge · Answer

It seems as if $$\Large x_n =a^{F_n}b^{F_{n+1}}c^{F_n}$$Where $F_n$ is the $n$-th Fibonacci number.

kinggeorge · Answer

Nevermind. I screwed up on the power of c. One minute.

kinggeorge · Answer

I'd like to correct it to be $$\Large x_n =a^{F_n}b^{F_{n+1}}c^{F_{n+2}-1}$$

anonymous · Answer

http://www.wolframalpha.com/input/?i=RSolve%5B%7Ba%5Bn+%2B+1%5D+%3D%3D+c+a%5Bn+-+1%5D+a%5Bn%5D%2C+a%5B0%5D+%3D%3D+a%2C+a%5B1%5D+%3D+b%7D%2C+a%5Bn%5D%2C+n%5D

kinggeorge · Answer

Let's add an "for $n\geq 2$ then :)

blockcolder · Answer

Lol. The answer was supposed to be
$$\Large x_n=a^{F_{n-2}}b^{F_{n-1}}c^{F_n-1}$$
and with the note that $F_{-2}=1$ and $F_{-1}=0$ but that works too.
@FoolForMath I don't know what a Lucas number is. o_O

kinggeorge · Answer

A Lucas number is defined as a recursive sequence where $L_0=2$, $L_1=1$ and $L_n = L_{n-1}+L_{n-2}$ for all other positive integers $n$.