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A002086
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Number of circulant tournaments on 2n+1 nodes up to Cayley isomorphism.
(Formerly M0939 N0353)
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4
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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
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OFFSET
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1,3
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REFERENCES
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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).
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LINKS
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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).
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MATHEMATICA
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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]];
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Roderick J. Fletcher, Oct 15 1996 (yylee(AT)mail.ncku.edu.tw)
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STATUS
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approved
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