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A002086 Number of circulant tournaments on 2n+1 nodes up to Cayley isomorphism.
(Formerly M0939 N0353)
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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

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-Fran├žois Alcover, Jul 02 2018, after Andrew Howroyd *)

PROG

(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

CROSSREFS

Cf. A002087, A049288.

Sequence in context: A206987 A207845 A207808 * A039830 A159788 A179112

Adjacent sequences:  A002083 A002084 A002085 * A002087 A002088 A002089

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

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

STATUS

approved

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Last modified March 20 07:42 EDT 2019. Contains 321345 sequences. (Running on oeis4.)