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

%I A002086 M0939 N0353 #38 Oct 14 2023 21:11:08
%S A002086 1,1,2,4,4,6,16,16,30,88,94,208,472,586,1096,3280,5472,7286,21856,
%T A002086 26216,49940,175104,182362,399480,1048576,1290556,3355456,7456600,
%U A002086 9256396,17895736,59660288,89478656,130150588,390451576,490853416,954437292,3435974656
%N A002086 Number of circulant tournaments on 2n+1 nodes up to Cayley isomorphism.
%D A002086 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002086 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002086 Andrew Howroyd, <a href="/A002086/b002086.txt">Table of n, a(n) for n = 1..200</a>
%H A002086 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 A002086 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 A002086 <a href="/index/To#tournament">Index entries for sequences related to tournaments</a>
%t A002086 IsLeastPoint[s_, f_] := Module[{t = f[s]}, While[t > s, t = f[t]]; s == t];
%t A002086 C0[n_, k_] := Sum[Boole @ IsLeastPoint[u, Mod[#*k, n]&], {u, 1, n-1}]/2;
%t A002086 IsBidrected[s_, r_, f_] := Module[{t = f[s]}, While[t != s && t != r, t = f[t]]; t == r];
%t A002086 IsC[n_, k_] := Sum[Boole @ IsBidrected[u, n-u, Mod[#*k, n]&], {u, 1, n-1}] == 0;
%t A002086 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 A002086 Array[a, 40] (* _Jean-François Alcover_, Jul 02 2018, after _Andrew Howroyd_ *)
%o A002086 (PARI)
%o A002086 IsLeastPoint(s,f)={my(t=f(s));while(t>s,t=f(t));s==t}
%o A002086 C(n,k)=sum(u=1,n-1,IsLeastPoint(u,v->v*k%n))/2;
%o A002086 IsBidrected(s,r,f)={my(t=f(s));while(t<>s&&t<>r,t=f(t));t==r}
%o A002086 IsC(n,k)=sum(u=1,n-1,IsBidrected(u,n-u,v->v*k%n))==0;
%o A002086 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 A002086 Cf. A002087, A049288.
%K A002086 nonn
%O A002086 1,3
%A A002086 _N. J. A. Sloane_
%E A002086 More terms from Roderick J. Fletcher, Oct 15 1996 (yylee(AT)mail.ncku.edu.tw)
%E A002086 Definition corrected by _Andrew Howroyd_, Apr 28 2017
%E A002086 Terms a(32) and beyond from _Andrew Howroyd_, Sep 30 2017