[seqfan] Re: A potentially new sequence based on a simple game

[Peter Pein]

Andrew Plewe schrieb:
> I have a game for the list; I'd be somewhat surprised if it hasn't  
> been "solved" before as it's a pretty simple game. Let set A be the  
> set of odd numbers starting with 3:
> 
> {3,5,7,etc..}
> 
> When you form a sum table, you get the even numbers starting with 6,  
> or set B:
> 
> {6,8,10,etc..}
> 
> Like this:
> 
>     3  5   7
> 3  6  8   10
> 5     10  12
> 7         14
> 
> The "complete" expression of B shown here is {6,8,10,12,14}; in other  
> words, when you take the first 3 characters from set A,  {3,5,7},  and  
> use that subset to form a sum table you get the first 5 characters of  
> set B. The game is, can we remove any characters from a subset of A  
> and still get a "complete" expression of  set B? In this case you  
> can't; removing 3 or 5 or 7 would render the resulting sum table  
> incomplete.
> 
> I believe the first subset of A that allows for a deletion is  
> {3,5,7,9,11}, which in turn "expresses" {6,8,10,12,14,16,18,20,22}.  
> You can remove 7 and you'll still get all of those values when you  
> compute the sum table:
> 
>      3   5   9   11
> 3   6   8   12  14
> 5       10  14  16
> 9           18  20
> 11              22
> 
> Now, the sequence is formed by counting the maximum number of sums  
> that can be removed from the sum table when you play this game for a  
> subset of A of length n. Here are the first few values of the sequence  
> (based on by-hand computation), starting with n = 1:
> 
> 0,0,0,0,5,6,13,15,22
> 
> and the corresponding subsets of A are:
> {3}
> {3,5}
> {3,5,7}
> {3,5,7,9}
> {3,5,9,11}
> {3,5,7,11,13}  or  {3,5,9,11,13}
> {3,5,9,13,15}
> {3,5,9,13,15,17}
> {3,5,9,13,17,19}
> 
> Thus in the example above n = 5 and when we removed the number 7 from  
> the subset we removed 5 sums from the resulting sum table. In other  
> words, imagine writing out the sum table for {3,5,7,9,11} and deleting  
> the row and the column starting with 7 and counting the deleted sums.
> 
> Is there an algorithmic way to determine this number? It would appear  
> to be so (I think there's a fairly simple pattern, but I'm not quite  
> sure about it yet), but if that's the case then odd things happen if  
> you let your subset equal A and thus extend to infinity.
> 
> 	-Andrew Plewe-
> 

Hi Andrew,

It would be nice if there was a _simple_ pattern, but I can't find one.
Using a brute-force backtracking method [1] I get the following start of this
sequence:

0,0,0,0,5,6,13,15,24,27,38,42,55,69,75,91,108,116,135,155,176,187,210,222,247...

which differs at a(9)=24 from your value 22 (maybe a typo?). This value is
reached when you take {7, 11, 15} out of A_9 ={3,5,...,19}.

Peter.

[1] http://home.arcor.de/petsie/seqfans/AndrewsGame.nb (Mathematica notebook)
and http://home.arcor.de/petsie/seqfans/AndrewsGame.pdf (the same as pdf )