A002086 Number of circulant tournaments on 2n+1 nodes up to Cayley isomorphism.

1, 1, 2, 4, 4, 6, 16, 16, 30, 88, 94, 208, 472, 586, 1096, 3280, 5472, 7286, 21856, 26216, 49940, 175104, 182362, 399480, 1048576, 1290556, 3355456, 7456600, 9256396, 17895736, 59660288, 89478656, 130150588, 390451576, 490853416, 954437292, 3435974656

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• 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

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• B. Alspach, On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323. See g(n) as defined on page 322 (NOT on page 317).
• B. Alspach, On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323. [Annotated copy] See g(n) as defined on page 322 (NOT on page 317).
• Index entries for sequences related to tournaments

Crossrefs

Cf. A002087, A049288.

Programs

Mathematica

IsLeastPoint[s_, f_] := Module[{t = f[s]}, While[t > s, t = f[t]]; s == t];
C0[n_, k_] := Sum[Boole @ IsLeastPoint[u, Mod[#*k, n]&], {u, 1, n-1}]/2;
IsBidrected[s_, r_, f_] := Module[{t = f[s]}, While[t != s && t != r, t = f[t]]; t == r];
IsC[n_, k_] := Sum[Boole @ IsBidrected[u, n-u, Mod[#*k, n]&], {u, 1, n-1}] == 0;
a[n_] := Module[{m = 2*n + 1}, Sum[If [GCD[m, k] == 1 && IsC[m, k], 2^C0[m, k], 0], {k, 1, m}]/EulerPhi[m]];
Array[a, 40] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

PARI

IsLeastPoint(s,f)={my(t=f(s));while(t>s,t=f(t));s==t}
C(n,k)=sum(u=1,n-1,IsLeastPoint(u,v->v*k%n))/2;
IsBidrected(s,r,f)={my(t=f(s));while(t<>s&&t<>r,t=f(t));t==r}
IsC(n,k)=sum(u=1,n-1,IsBidrected(u,n-u,v->v*k%n))==0;
a(n)=my(m=2*n+1);sum(k=1, m, if (gcd(m,k)==1 && IsC(m,k), 2^C(m,k),0))/eulerphi(m); \\ Andrew Howroyd, Sep 30 2017

Extensions

More terms from Roderick J. Fletcher, Oct 15 1996 (yylee(AT)mail.ncku.edu.tw)

Definition corrected by Andrew Howroyd, Apr 28 2017

Terms a(32) and beyond from Andrew Howroyd, Sep 30 2017