A047656 a(n) = 3^((n^2-n)/2).

1, 1, 3, 27, 729, 59049, 14348907, 10460353203, 22876792454961, 150094635296999121, 2954312706550833698643, 174449211009120179071170507, 30903154382632612361920641803529, 16423203268260658146231467800709255289, 26183890704263137277674192438430182020124347

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The number of outcomes of a chess tournament with n players.

For n >= 1, a(n) is the size of the Sylow 3-subgroup of the Chevalley group A_n(3) (sequence A053290). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001

The number of binary relations on an n-element set that are both reflexive and antisymmetric. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005

The sequence a(n+1) = [1,3,27,729,59049,14348907,...] is the Hankel transform (see A001906 for definition) of A047891 = 1, 3, 12, 57, 300, 1586, 9912, ... . - Philippe Deléham, Aug 29 2006

a(n) is the number of binary relations on a set with n elements that are total relations, i.e., for a relation on a set X it holds for all a and b in X that a~b or b~a (or both). E.g., a(2) = 3 because there are three total relations on a set with two elements: {(a,a),(a,b),(b,a),(b,b)}, {(a,a),(a,b),(b,b)}, and {(a,a),(b,a),(b,b)}. - Geoffrey Critzer, May 23 2008

The number of semicomplete digraphs (or weak tournaments) on n labeled nodes. - Rémy-Robert Joseph, Nov 12 2012

The number of n X n binary matrices A that have a(i,j)=0 whenever a(j,i)=1 for i!=j and zeros on the diagonal. We need only consider the (n^2-n)/2 non-diagonal entry pairs . Since each pair is of the form <0,0>, <0,1>, or <1,0>, a(n) = 3^((n^2-n)/2). - Dennis P. Walsh, Apr 03 2014

a(n) is the number of symmetric (-1,0,1)-matrices of dimension (n-1) X (n-1). - Eric W. Weisstein, Jan 03 2021

Examples

			The a(2)=3 binary 2 X 2 matrices are [0 0; 0 0], [0 1; 0 0], and [0 0; 1 0]. - _Dennis P. Walsh_, Apr 03 2014
		

References

• P. A. MacMahon, Chess tournaments and the like treated by the calculus of symmetric functions, Coll. Papers I, MIT Press, 344-375.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..65
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
• T. R. Hoffman and J. P. Solazzo, Complex Two-Graphs via Equiangular Tight Frames, arXiv preprint arXiv:1408.0334 [math.CO], 2014.
• G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
• Eric Weisstein's World of Mathematics, (-1,0,1)-Matrix
• Eric Weisstein's World of Mathematics, Symmetric Matrix
• Index entries for sequences related to tournaments

Crossrefs

Cf. A007747.

Programs

Maple

seq(3^binomial(n, 2), n=0..12); # Zerinvary Lajos, Jun 16 2007
seq(3^((n^2-n)/2), n=0..14);

Mathematica

Table[3^((n^2 - n)/2), {n, 0, 14}] (* Eric W. Weisstein, Jan 03 2021 *)
3^Table[Binomial[n, 2], {n, 0, 14}] (* Eric W. Weisstein, Jan 03 2021 *)
3^Binomial[Range[0, 14], 2] (* Eric W. Weisstein, Jan 03 2021 *)
Table[Count[Tuples[{-1, 0, 1}, {n, n}], ?SymmetricMatrixQ], {n, 3}] (* _Eric W. Weisstein, Jan 03 2021 *)

PARI

a(n)=3^binomial(n+1,2) \\ Charles R Greathouse IV, Apr 17 2012

Formula

a(n+1) is the determinant of an n X n matrix M_(i, j) = C(3*i,j). - Benoit Cloitre, Aug 27 2003

Sequence is given by the Hankel transform (see A001906 for definition) of A007564 = {1, 1, 4, 19, 100, 562, 3304, ...}; example: det([1, 1, 4, 19; 1, 4, 19, 100; 4, 19, 100, 562; 19, 100, 562, 3304]) = 3^6 = 729. - Philippe Deléham, Aug 20 2005

The sequence a(n+1) = [1,3,27,729,59049,14348907,...] is the Hankel transform (see A001906 for definition) of A047891 = 1, 3, 12, 57, 300, 1586, 9912, ... . - Philippe Deléham, Aug 29 2006

a(n) = 3^binomial(n,2). - Zerinvary Lajos, Jun 16 2007

G.f. A(x) satisfies: A(x) = 1 + x * A(3*x). - Ilya Gutkovskiy, Jun 04 2020

a(n) = a(n-1)*3^(n-1), a(0) = 1. - Mehdi Naima, Mar 09 2022