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%I #18 Nov 03 2020 13:52:27
%S 1,0,2,0,6,0,20,2,70,18,252,110,924,572,3434,2730,12902,12376,48926,
%T 54264,187036,232562,720062,980674,2789164,4086550,10861060,16878420,
%U 42484682,69242082,166823430,282580872,657178982,1148548016,2595874468
%N Number of closed walks of length n at a vertex of the cyclic graph on 7 nodes C_7.
%C In general a(n,m)=2^n/m*Sum_{k=0..m-1} cos(2Pi*k/m)^n) counts closed walks of length n at a vertex of the cyclic graph on m nodes C_m.
%H Harvey P. Dale, <a href="/A094659/b094659.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1, 4, -3, -2).
%F a(n ) =2^n/7*Sum_{k=0..6} cos(2Pi*k/7)^n).
%F a(n) = 7(a(n-2) - 2a(n-4) + a(n-6)) + 2a(n-7).
%F G.f.: (1-x-2x^2+x^3)/((2x-1)(-1-x+2x^2+x^3)).
%F a(0)=1, a(1)=0, a(2)=2, a(3)=0, a(n)=a(n-1)+4*a(n-2)-3*a(n-3)-2*a(n-4). - _Harvey P. Dale_, Jun 12 2014
%F 7*a(n) = 2^n +2*A094648(n). - _R. J. Mathar_, Nov 03 2020
%t f[n_] := FullSimplify[ TrigToExp[ 2^n/7 Sum[Cos[2Pi*k/7]^n, {k, 0, 6}]]]; Table[ f[n], {n, 0, 36}] (* _Robert G. Wilson v_, Jun 09 2004 *)
%t LinearRecurrence[{1,4,-3,-2},{1,0,2,0},40] (* _Harvey P. Dale_, Jun 12 2014 *)
%K nonn,easy
%O 0,3
%A _Herbert Kociemba_, Jun 06 2004
%E More terms from _Robert G. Wilson v_, Jun 09 2004
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