wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> medium >> 1, 2, 6, 42, ...
        
(Message started by: cool_joh on Nov 22^nd, 2007, 8:31pm)

Title: 1, 2, 6, 42, ...
Post by cool_joh on Nov 22^nd, 2007, 8:31pm

a_n = a_n-1 (a_n-1 + 1)

How can we find the common formula for a_n?

Title: Re: 1, 2, 6, 42, ...
Post by towr on Nov 23^rd, 2007, 12:15am

If you mean a closed formula, that may be tricky. I can't find a general method for solving these things (see also http://mathworld.wolfram.com/QuadraticMap.html )

Title: Re: 1, 2, 6, 42, ...
Post by Barukh on Nov 23^rd, 2007, 1:11am

on 11/23/07 at 00:15:11, towr wrote:

I can't find a general method for solving these things

It's available here (http://www.research.att.com/~njas/doc/doubly.html).

I find the following very beautiful (taken from Sloane A000058 (http://www.research.att.com/~njas/sequences/A000058)):

"The greedy Egyptian representation of 1 is 1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + ... "

which is the sum of reciprocals of a_n + 1.

Title: Re: 1, 2, 6, 42, ...
Post by cool_joh on Nov 23^rd, 2007, 4:28am

on 11/23/07 at 01:11:46, Barukh wrote:

It's available here (http://www.research.att.com/~njas/doc/doubly.html).

It's too complicated. I'm not good in English, can someone help me please?

Title: Re: 1, 2, 6, 42, ...
Post by Barukh on Nov 23^rd, 2007, 9:08am

Bottom line is this: there exist a constant r, such that a_n equals to the nearest integer of r^{2^n} (sequence starts at n=0). For this particular sequence, r equals 1.5979...

That's kind of cheating since to compute r one needs to know all a_n-s upfront (formula (19) in the article). The series converges very rapidly, though.

Looks like it's the best that could be done (or is known).