Convert Recursive to Closed Formula

I have a sequence defined by the following recursive formula:

$\displaystyle T_n = T_{n-1} \times 2 - T_{n-10}$

This produces the sequence:

*n = 1 to 10:* 1, 1, 2, 4, 8, 16, 32, 64, 128, 256

*n = 11 to 15:* 511, 1021, 2040, 4076, 8144

note: n=1 is an anomaly.

I need help converting the formula to a closed form. The closed formula doesn't have to reproduce the above sequence for the n = 1 to 10 parts if it makes it any easier. I can account for those special cases in my program.

Re: Convert Recursive to Closed Formula

$\displaystyle 2^{n-2}-(n-9)2^{n-12}$?

Re: Convert Recursive to Closed Formula

That answer works from n = 11 to 20, but fails beyond that.

Re: Convert Recursive to Closed Formula

Sorry. I knew it couldn't be that easy really. The characteristic polynomial only has one root that's nice to work with:

1.998029470262286698....

Beyond a certain point your sequence is probably almost indistinguishable from a GP. Using some term, m say, of your sequence you can generate others using $\displaystyle T_n=$ nearest integer to $\displaystyle T_m*1.998.....^{n-m}$

I hope I'm not wrong again. I could check but I'm sure you will. (Sweating)

Any experts reading this thread?

Re: Convert Recursive to Closed Formula

@a tutor> Thanks for your help. After researching & trialing different methods, I gave up trying to find the closed form using **Linear Homogeneous Recurrence Relation **like you suggested. Instead, I am now using **Transitional Matrix**.

My original intent for finding the closed form was for efficiency for very large values of n, but I found that transitional matrix works surprisingly efficiently as well.