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A000198
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Largest order of automorphism group of a tournament with n nodes.
(Formerly M2280 N0902)
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1
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1, 1, 3, 3, 5, 9, 21, 21, 81, 81, 81, 243, 243, 441, 1215, 1701, 1701, 6561, 6561, 6561, 45927, 45927, 45927, 137781, 137781, 229635, 1594323, 1594323, 1594323, 4782969, 4782969, 7971615, 14348907, 33480783, 33480783, 129140163, 129140163, 129140163
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OFFSET
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1,3
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COMMENTS
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It appears that all terms except for a(5) = 5 are divisible by a power of 3. - Jonathan Vos Post, Apr 20 2011
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REFERENCES
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J. W. Moon, Topics on Tournaments. Holt, NY, 1968, p. 81.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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B. Alspach, On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323. [Annotated copy] See g(n) as defined on page 317 (NOT on page 322).
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FORMULA
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a(3^k) = 3^((3^k - 1)/2), a(5*3^k) = 5*3^((5*3^k - 5)/2), a(7*3^k) = 7*3^((7*3^k - 5)/2), and, for all other n, a(n) = max(a(i)a(n-i)) where the maximum is taken over 1 <= i <= n-1 (from Alspach and Berggren (1973) Theorem 4).
a(3r) = (3^r)a(r), a(n) = a(n-1) for n = 1, 2 or 4 mod 9, a(9k+8) = max(a(9k+7), a(5)a(9k+3)), a(9k+5) = max(a(2)a(9k+3), a(5)a(9k), a(7)a(9k-2)), a(9k+7) = a(7)a(9k) (from Alspach and Berggren (1973) Theorem 5).
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MAPLE
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a:= proc(n) local t, r; t:= irem(n, 9);
`if`(3^ilog[3](n)=n, 3^((3^ilog[3](n)-1)/2),
`if`(irem(n, 5, 'r')=0 and 3^ilog[3](r)=r, 5*3^((5*3^ilog[3](r)-5)/2),
`if`(irem(n, 7, 'r')=0 and 3^ilog[3](r)=r, 7*3^((7*3^ilog[3](r)-5)/2),
`if`(irem(n, 3, 'r')=0, 3^r*a(r),
`if`(t in {1, 2, 4}, a(n-1),
`if`(t = 8, max(a(n-1), a(5)*a(n-5)),
`if`(t = 5, max(a(2)*a(n-2), a(5)*a(n-5), a(7)*a(n-7)),
a(7)*a(n-7) )))))))
end:
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MATHEMATICA
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a[n_] := a[n] = With[{t = Mod[n, 9]}, Which[ IntegerQ[Log[3, n]], 3^((1/2)*(n-1)), {q, r} = QuotientRemainder[n, 5]; r == 0 && IntegerQ[Log[3, q]], 5*3^((1/2)*(n-5)), {q, r} = QuotientRemainder[n, 7]; r == 0 && IntegerQ[Log[3, q]], 7*3^((1/2)*(n-5)), {q, r} = QuotientRemainder[n, 3]; r == 0, 3^q*a[q], MemberQ[{1, 2, 4}, t], a[n-1], t == 8, Max[a[n-1], a[5]*a[n-5]], t == 5, Max[a[2]*a[n-2], a[5]*a[n-5], a[7]*a[n-7]], True, a[7]*a[n-7]]]; Table[a[n], {n, 1, 38}] (* Jean-François Alcover, Nov 12 2012, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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