[seqfan] Re: game series

Peter Pein petsie at dordos.net
Fri Jan 28 13:21:35 CET 2011


On 28.01.2011 10:10, Dmitry Kamenetsky wrote:
> Hello Sequence Fans,
>
> Recently I have been looking at the following problem. There is a series of
> n games played between two teams. The outcome of each game is either a win
> or a loss (no draws).
> A team wins the whole series if it wins k=floor(n/2)+1 games or more. Now if
> a team reaches the magic number of k wins then the games that follow (if
> there are any) are
> dead games, because their outcome cannot affect the outcome of the series.
> So a natural question arises: out of all the possible 2^n series how many of
> them will have
> at least one dead game? This forms the sequence 0,0,4,4,20,24,88,116,372,...
> This sequence is not in the OEIS and neither is its version for all odd n.
Hi Dimitry,

what did I do wrong while trying to reconstruct the sequence 
{0,0,4,4,20,24,88..} ?

I tried to find those sequences of wins/losses which contain a sequence 
of wins/losses of length >= Floor[n/2]+1 followed by at least one "dead" 
game:

In[1]:= 
f[n_]:=Count[Tuples[{0,1},{n}],({___,0,s:1..,0,__}|{___,1,s:0..,1,__})/;Length[{s}]>=Floor[n/2]+1]
In[3]:= f/@Range[20]
Out[3]= {0,0,0,0,0,0,4,4,20,20,68,68,196,196,516,516,1284,1284,3076,3076}

Did I misunderstand your definition? Please enlighten me!

Cheers,
   Peter




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