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Connell Sequence

Date: 06/12/2003 at 23:40:49
From: Rachelle
Subject: Number sequence

The following sequence

1,2,4,5,7,9,10,12,14,16,17...

has one odd number followed by two evens, then three odds, four evens, 
and so on. What number is the 2003th term?


Date: 06/16/2003 at 14:19:58
From: Doctor Douglas
Subject: Re: Number sequence

Hi Rachelle,

For this problem, it's a lot easier to see what happens if we format 
the sequence in a triangle:

    1   2   5  10  17 . . .
    4   7  12  19
    9  14  21
   16  23
   25
    .
    .
    .

You can see that each diagonal is either all even or all odd, and that 
they alternate. Also, notice that the numbers in the first column are 
all perfect squares (this observation will save us a lot of algebraic 
work).

Suppose we reach the number k^2 in the sequence. This will happen at 
the kth triangular number, i.e. at at index of k(k+1)/2. For example, 
25 = 5 x 5 has k = 5, so it is the 5(5+1)/2 = 15th number in the 
sequence.  

We need the smallest triangular number bigger than 2003. We can 
estimate the k for which this happens by considering sqrt(2003*2) =
sqrt(4006) = 63.29.

  63 x 64/2 = 2016
  62 x 63/2 = 1953

This tells us that the 2003rd number in the sequence is an ODD number 
somewhere between 62^2 = 3844 and 63^2 = 3969. We know that 3969 is 
the 2016th number in the sequence. Since we want the 2003rd, we know 
that it is 13 members before 3969, or

  3969 - 13 x 2 = 3969 - 26 = 3943.

Incidentally, this sequence is sometimes known as the "Connell
sequence."  There is even an explicit formula for the nth term:

  a(n) = 2n - Floor[(1 + Sqrt(8n - 7))/2]

The notation "Floor(x)" means take x and round down to the next 
integer. If you plug n = 2003 into this, you'll obtain the answer that 
we found above.  

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Sequences, Series

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