The definition of the sequence is colloquially described as:
* &nbsp; Starting with the list [1,1],
* &nbsp; Take the last number in the list so far: 1, I'll call it x.
* &nbsp; Count forward x places from the beginning of the list to find the first number to add (1)
* &nbsp; Count backward x places from the end of the list to find the second number to add (1)
* &nbsp; Add the two indexed numbers from the list and the result becomes the next number in the list (1+1)
* &nbsp; This would then produce [1,1,2] where 2 is the third element of the sequence.

<br>Note that indexing for the description above starts from alternately the left and right ends of the list and starts from an index of ''one''.

A less wordy description of the sequence is:
    a(1)=a(2)=1
    a(n)=a(a(n-1))+a(n-a(n-1))

The sequence begins:
    1, 1, 2, 2, 3, 4, 4, 4, 5, ...

Interesting features of the sequence are that:
* &nbsp; a(n)/n &nbsp; tends to &nbsp; 0.5 &nbsp; as &nbsp; n &nbsp; grows towards infinity.
* &nbsp; a(n)/n &nbsp; where &nbsp; n &nbsp; is a power of &nbsp; 2 &nbsp; is &nbsp; 0.5
* &nbsp; For &nbsp; n>4 &nbsp; the maximal value of &nbsp; a(n)/n &nbsp; between successive powers of 2 decreases.

[[File:Hofstadter conway 10K.gif|center|a(n) / n &nbsp; for &nbsp; n &nbsp; in &nbsp; 1..256]] 


The sequence is so named because [[wp:John Horton Conway|John Conway]] [http://www.nytimes.com/1988/08/30/science/intellectual-duel-brash-challenge-swift-response.html offered a prize] of $10,000 to the first person who could
find the first position, &nbsp; p &nbsp; in the sequence where 
    │a(n)/n│ < 0.55  for all  n > p
It was later found that [[wp:Douglas Hofstadter|Hofstadter]] had also done prior work on the sequence.

The 'prize' was won quite quickly by [https://en.wikipedia.org/wiki/Colin_Lingwood_Mallows Dr. Colin L. Mallows] who proved the properties of the sequence and allowed him to find the value of &nbsp; n &nbsp; (which is much smaller than the 3,173,375,556 quoted in the NYT article).


;Task:
# &nbsp; Create a routine to generate members of the Hofstadter-Conway $10,000 sequence.
# &nbsp; Use it to show the maxima of &nbsp; a(n)/n &nbsp; between successive powers of two up to &nbsp; 2**20
# &nbsp; As a stretch goal: &nbsp; compute the value of &nbsp; n &nbsp; that would have won the prize and confirm it is true for &nbsp; n &nbsp; up to 2**20


<br>
;Also see:
* &nbsp; [http://www.jstor.org/stable/2324028 Conways Challenge Sequence], Mallows' own account.
* &nbsp; [http://mathworld.wolfram.com/Hofstadter-Conway10000-DollarSequence.html Mathworld Article].
<br><br>

