A000198 Largest order of automorphism group of a tournament with n nodes. (Formerly M2280 N0902)

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

Offset

1,3

References

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).

Links

Joseph Myers, Table of n, a(n) for n = 1..1000

B. Alspach, A combinatorial proof of a conjecture of Goldberg and Moon, Canad. Math. Bull. 11 (1968), 655-661.

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).

B. Alspach and J. L. Berggren, On the determination of the maximum order of the group of a tournament, Canad. Math. Bull 16 (1973), 11-14.

J. D. Dixon, The maximum order of the group of a tournament, Canad. Math. Bull. 10 (1967), 503-505.

Index entries for sequences related to tournaments

Formula

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).

Maple

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:
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 29 2012

Mathematica

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 *)

Prog

(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A000198(n):
    if n <= 7: return (1, 1, 3, 3, 5, 9, 21)[n-1]
    if (r:=n%9) in {0, 3, 6}:
        return 3**(m:=n//3)*A000198(m)
    elif r in {1, 2, 4}:
        return A000198(n-1)
    elif r == 5:
        return max(A000198(n-2), 5*A000198(n-5), 21*A000198(n-7))
    elif r == 7:
        return 21*A000198(n-7)
    elif r == 8:
        return max(A000198(n-1), 5*A000198(n-5)) # Chai Wah Wu, Jul 01 2024

Crossrefs

Cf. A000568, A374164.

Sequence in context: A209083 A137202 A146926 * A202674 A027170 A132775

Adjacent sequences: A000195 A000196 A000197 * A000199 A000200 A000201

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Edited and extended by Joseph Myers, Jun 28 2012

Status

approved