Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is there a closed form of the following sequence:

$$u_0 = 2$$

$$u_{n+1} = s_n^2-s_n, \;s_n = \sum_{k=0}^{n} u_k$$

If not, I would like to have an upper bound. By looking at the numbers I guessed that $2^{2^n}$ is one, is this true? Is there a better one?

share|improve this question

1 Answer 1

up vote 3 down vote accepted

$u_{n+1}=s_{n+1}-s_n=s_n^2-s_n$, therefore $s_{n+1}=s_n^2$. Hence $\log(s_{n+1})=2\log(s_n)$. Let $b_n=\log(s_n)$. Solve for $b_n$ and then $s_n$. Then find $u_n$:

$$u_n=s_n-s_{n-1}$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.