# [seqfan] Re: question on a recursively defined sequence

Jim Nastos nastos at gmail.com
Mon Feb 23 01:29:13 CET 2009

```Sorry, a little typo:

The written line :
" Then t(n) = 3 + SUM (i=1 to n-2) PROD (j=1 to i)  2Tower(i,3)  "

should be corrected to " 2Tower(j,3)  "

(in case someone wanted to blindly implement that.)

Sorry for the many messages.
JN

On Sun, Feb 22, 2009 at 4:26 PM, Jim Nastos <nastos at gmail.com> wrote:
> So, once the recursion is written with a single recursive call,
> iteration will lead to a "closed form" although
> it isn't pretty. In involves towers-of-powers.
>
> t(1) = 1, t(2) = 3
> t(3) = 3 + 2^3
> t(4) = 3 + 2^3 + 2^3 2^(2^3)
> t(5) = 3 + 2^3 + 2^3 2^(2^3) + 2^3 2^(2^3) 2^(2^(2^3))
>
> So, use the function 2Tower(k,N) to mean 2^...^(2^N) where there are k
> "2"s written in the tower.
>
> Then t(n) = 3 + SUM (i=1 to n-2) PROD (j=1 to i)  2Tower(i,3)
>
> Tower functions rarely have a way of being expressed without a big-Pi
> (product notation.) Most
> understandings of "closed form" mean without big-Sigma or big-Pi
> notation, so although this gives
> a non-recursive formula to calculate the n-th term, it isn't really a
> closed form.
>
> I'll agree with Dr. McKay that a closed form is unlikely.
>
> JN
>
>
>
> On Sun, Feb 22, 2009 at 4:04 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?
>>
>> JN
>>
>>
>> On Sun, Feb 22, 2009 at 3:51 PM, T. D. Noe <noe at sspectra.com> wrote:
>>> At 3:44 PM -0800 2/22/09, Jonathan Post wrote:
>>>>It's on that page thus:
>>>>
>>>>Closed form of a recursive series
>>>>
>>>>Is there a closed form with a fixed number of terms for the sequence
>>>>defined by t0 = 1 and tn = tn - 1 + 2tn - 1? In case anyone's
>>>>wondering, it originates here. NeonMerlin 08:11, 22 February 2009
>>>>(UTC)
>>>>
>>>>    I doubt there's anything useful. There's nothing in the Sloane's
>>>>entry. Algebraist 09:40, 22 February 2009 (UTC)
>>>>
>>>>On Sun, Feb 22, 2009 at 3:34 PM, Brendan McKay <bdm at cs.anu.edu.au> wrote:
>>>>> At the Mathematics Reference Desk of Wikipedia
>>>>>   http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Mathematics
>>>>> someone is asking for a closed form for t[n] where
>>>>>  t = 1, t[n] = t[n-1] + 2^(t[n-1]).
>>>>> It looks unlikely to me.  Does anyone disagree?
>>>
>>>
>>> Sequence A034797: http://www.research.att.com/~njas/sequences/A034797
>>>
>>> Tony
>>>
>>>
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>
>

```