[seqfan] Re: question on a recursively defined sequence

[Max Alekseyev]

On Sun, Feb 22, 2009 at 7:04 PM, Jim Nastos <nastos at gmail.com> wrote:
> Telescoping once to remove the linear term gives a nicer recurrence
> (only makes one recursive call
> instead of two.) I'm surprised it's not written in the OEIS page.
>
> t(1) = 1
> t(2) = 3
> t(n) = 3 + 2^t(n-1)
>
> Can someone get a closed form from this?

Let s(n)=(t(n)-3)/8 then

s(n) = 256^s(n-1)

The closed formula for s(n) is
s(n) = 256^256^256^...^256^1 for n>=3.
where there are n-3 exponentiations in the tower.

Clearly, t(n) = 8*s(n) + 3.

Regards,
Max