What are the next two numbers in this sequence?
1, 1, 7, 3, 23, 23, 111, 15, 351, 415, __, __

I guess 1791 and 255

If the guess is correct, I have no idea how you derived the sequence.  But - here's what I have done.....  This is a total patternistic approach to solving your equation - but none-the-less what I attempted.... Since I certainly haven't solved the equation, this is more FYI of what I did.

Taking the original sequence, find the Ä to each additional #:

#      Ä          k

0 (assume)   
1        1         0
1        0         1
7        6         2
3       -4        3
23      20       4
23      0         5
111    88        6
15     -96       7
351    336      8
415    64        9
___    ____    10
___    ____    11

By pulling all the even Ä's:  1,6,20,88,336,___

You can derive an equation:  (Yes, this only works for the even Ä's)

                                      (k/2+1)         (k/2)
ÄA(k) =  4 * ÄA(k-2) + (-1           ) * (2        )

Yielding:
ÄA(0) = 1 (By definition of my assumption)
ÄA(2) =  4 * 1 + 1 * 2 = 6
ÄA(4) =  4 * 6 + -1 * 4 = 20
ÄA(6) =  4 * 20 + 1 * 8 = 88
ÄA(8) =  4 * 88 + -1 * 16 = 336
All of these are correct Ä's according to true numbers..... so
ÄA(10) = 4 * 336 + 1 * 32 = 1376

Now - knowing that ÄA(10) = 1376, that makes the next number in the series 415 + 1376 = 1791

As for the last number....  this is truly just a patterned guess for me....
Looking at the Binary - shown in Danny's post, the 4th number, 3, is 2 ones.  The 8th number, 15, is 4 ones, corresponding also to the last 4 ones of the 7th number, 111.  Therefore I predict / guess the last number to be 8 ones, also corresponding to the last 8 ones of the 11th number.  Hence my guess of 255.

Well - looking forward to the real answers and real equation of the sequence - and still stumped!

Posted by Telly on 2004-06-09 18:46:49