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%I #22 Sep 08 2022 08:45:25
%S 2,6,84,1020,15630,279930,5764808,134217720,3486784410,99999999990,
%T 3138428376732,106993205379060,3937376385699302,155568095557812210,
%U 6568408355712890640,295147905179352825840,14063084452067724991026
%N Number of functions f: {1, 2, ..., n} --> {1, 2, ..., n} such that f(1) != f(2), f(2) != f(3), ..., f(n-1) != f(n), f(n) != f(1).
%C a(n) is also the number of circuits of length n in the complete graph on n vertices. - Thibaut Lienart (syncthib(AT)gmail.com), Jan 29 2010
%C Circuits are allowed to be self-intersecting and are directional with a designated start node. The number of (self-avoiding) directed cycles is given by A124355. - _Andrew Howroyd_, Sep 05 2018
%C a(n) is also the number of graph colorings of the cycle graph C_n with n colors. - _Orson R. L. Peters_, Jul 27 2020
%H Andrew Howroyd, <a href="/A118537/b118537.txt">Table of n, a(n) for n = 2..100</a>
%F a(n) = (n-1)^n + (-1)^n*(n-1).
%t a[n_]:=(n-1)^n + (-1)^n*(n-1); Array[a, 50, {2, 51}] (* _Stefano Spezia_, Sep 07 2018 *)
%o (PARI) a(n) = (n-1)^n + (-1)^n*(n-1); \\ _Andrew Howroyd_, Sep 05 2018
%o (Magma) [(n-1)^n + (-1)^n*(n-1) : n in [2..20]]; // _Wesley Ivan Hurt_, Jul 27 2020
%Y Cf. A055897, A124355.
%K nonn
%O 2,1
%A _Warut Roonguthai_, May 06 2006
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