This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For 2 teams, [0,3] is the only possible outcome with no repeated point totals, so a(2) = 1. For 3 teams, the only outcomes are [0,3,6], [1,2,4] and [1,3,4], so a(3) = 3.
With 3 players the possible scores sequences are {{0,2,4}, {0,3,3}, {1,1,4}, {1,2,3}, {2,2,2}}.
With 4 players they are {{0,2,4,6}, {0,2,5,5}, {0,3,3,6}, {0,3,4,5}, {0,4,4,4}, {1,1,4,6}, {1,1,5,5}, {1,2,3,6}, {1,2,4,5}, {1,3,3,5}, {1,3,4,4}, {2,2,2,6}, {2,2,3,5}, {2,2,4,4}, {2,3,3,4}, {3,3,3,3}}.
f[K_, L_, S_, X_] /; K > 1 && L <= S/K <= X + 1 - K := f[K, L, S, X] = Sum[f[K - 1, i, S - i, X], {i, L, Floor[S/K]}]; f[1, L_, S_, X_] /; L <= S <= X = 1; f[, , , ] = 0; a[n_] := f[n + 1, 0, n*(n + 1), 2*n]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 13 2012, after Jon E. Schoenfield *)
For 2 teams there are 4 possible outcomes: [0, 6], [1, 4], [2, 2] and [3, 3], so a(2) = 4.
a(1)..a(4) are the same as in A064626.
def play(ps, n, r, i, j):
if j>=n:
ps.add(tuple(sorted(r)))
else:
(ni,nj) = (i,j+1) if j<(n-1) else (i+1,i+2)
s=list(r)
s[i]=r[i]+n; play(ps,n,s,ni,nj)
s[i]=r[i]+1; s[j]=r[j]+1; play(ps,n,s,ni,nj)
s[i]=r[i] ; s[j]=r[j]+n; play(ps,n,s,ni,nj)
def A317723(n):
ps=set()
play(ps,n,[0]*n,0,1)
return len(ps)
# Bert Dobbelaere, Oct 07 2018
For n=2 the possible outcomes are [0,6] and [1,4], so a(2)=2. The other possible outcomes [2,2] and [3,3] do not have distinct point totals.
For 2 teams there are 2 outcomes that can be obtained in a single way: [0, 3] and [1, 1], so a(2) = 2. For 3 teams there are 6 outcomes that can be obtained in a single way: [0, 3, 6], [1, 3, 4], [1, 1, 6], [1, 2, 4], [0, 4, 4] and [2, 2, 2], so a(3) is 6. Note that the outcome [3, 3, 3] can be obtained in two ways: (A beats B, B beats C, C beats A) or (B beats A, A beats C, C beats B).
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