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Round-robin tournament numbers: The number of possible point series for a tournament of n teams playing each other once where n points are awarded to the winning team and 1 to each in the case of a tie. A team winning more games than another always has a higher point score.
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%I #32 Sep 16 2024 12:49:06

%S 1,2,7,40,367,4828,82788,1750152

%N Round-robin tournament numbers: The number of possible point series for a tournament of n teams playing each other once where n points are awarded to the winning team and 1 to each in the case of a tie. A team winning more games than another always has a higher point score.

%C The 3-point rule is equivalent to that for football (A064626).

%C The classical 2-point rule is equivalent to that for chess tournaments (A007747).

%H Donghwi Park, <a href="https://gist.github.com/gwahak/8e96bfe2438679aca0af7c33750916cc">source code for a(5)</a>

%e a(1)..a(4) are the same as in A064626.

%o (Python)

%o def play(ps, n, r, i, j):

%o if j>=n:

%o ps.add(tuple(sorted(r)))

%o else:

%o (ni,nj) = (i,j+1) if j<(n-1) else (i+1,i+2)

%o s=list(r)

%o s[i]=r[i]+n; play(ps,n,s,ni,nj)

%o s[i]=r[i]+1; s[j]=r[j]+1; play(ps,n,s,ni,nj)

%o s[i]=r[i] ; s[j]=r[j]+n; play(ps,n,s,ni,nj)

%o def A317723(n):

%o ps=set()

%o play(ps,n,[0]*n,0,1)

%o return len(ps)

%o # _Bert Dobbelaere_, Oct 07 2018

%Y Cf. A007747, A064626.

%K nonn,more,hard

%O 1,2

%A _Donghwi Park_, Aug 05 2018

%E a(6)-a(8) from _Bert Dobbelaere_, Oct 07 2018