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Number of circulant tournaments on 2n+1 nodes up to Cayley isomorphism.
(Formerly M0939 N0353)
4

%I M0939 N0353 #38 Oct 14 2023 21:11:08

%S 1,1,2,4,4,6,16,16,30,88,94,208,472,586,1096,3280,5472,7286,21856,

%T 26216,49940,175104,182362,399480,1048576,1290556,3355456,7456600,

%U 9256396,17895736,59660288,89478656,130150588,390451576,490853416,954437292,3435974656

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Andrew Howroyd, <a href="/A002086/b002086.txt">Table of n, a(n) for n = 1..200</a>

%H B. Alspach, <a href="http://dx.doi.org/10.4153/CMB-1970-061-7">On point-symmetric tournaments</a>, Canad. Math. Bull., 13 (1970), 317-323. See g(n) as defined on page 322 (NOT on page 317).

%H B. Alspach, <a href="/A002086/a002086.pdf">On point-symmetric tournaments</a>, Canad. Math. Bull., 13 (1970), 317-323. [Annotated copy] See g(n) as defined on page 322 (NOT on page 317).

%H <a href="/index/To#tournament">Index entries for sequences related to tournaments</a>

%t IsLeastPoint[s_, f_] := Module[{t = f[s]}, While[t > s, t = f[t]]; s == t];

%t C0[n_, k_] := Sum[Boole @ IsLeastPoint[u, Mod[#*k, n]&], {u, 1, n-1}]/2;

%t IsBidrected[s_, r_, f_] := Module[{t = f[s]}, While[t != s && t != r, t = f[t]]; t == r];

%t IsC[n_, k_] := Sum[Boole @ IsBidrected[u, n-u, Mod[#*k, n]&], {u, 1, n-1}] == 0;

%t 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]];

%t Array[a, 40] (* _Jean-François Alcover_, Jul 02 2018, after _Andrew Howroyd_ *)

%o (PARI)

%o IsLeastPoint(s,f)={my(t=f(s));while(t>s,t=f(t));s==t}

%o C(n,k)=sum(u=1,n-1,IsLeastPoint(u,v->v*k%n))/2;

%o IsBidrected(s,r,f)={my(t=f(s));while(t<>s&&t<>r,t=f(t));t==r}

%o IsC(n,k)=sum(u=1,n-1,IsBidrected(u,n-u,v->v*k%n))==0;

%o 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

%Y Cf. A002087, A049288.

%K nonn

%O 1,3

%A _N. J. A. Sloane_

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

%E Definition corrected by _Andrew Howroyd_, Apr 28 2017

%E Terms a(32) and beyond from _Andrew Howroyd_, Sep 30 2017