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%I #19 Mar 31 2021 04:10:03
%S 0,30,156,826,4406,23562,126104,675074,3614142,19349430,103593804,
%T 554625898,2969386478,15897666066,85113810056,455687062274,
%U 2439682811478,13061709929934,69930511268508,374397872321626
%N Number of ways for each of 2n (labeled) people in a circle to look at either a neighbor or the diametrally opposite person, such that no eye contact occurs.
%H G. C. Greubel, <a href="/A141221/b141221.txt">Table of n, a(n) for n = 1..1000</a>
%H Max A. Alekseyev and Gérard P. Michon, <a href="http://arxiv.org/abs/1602.01396">Making Walks Count: From Silent Circles to Hamiltonian Cycles</a>, arXiv:1602.01396 [math.CO], 2016.
%H Art of Problem Solving Forum, <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?t=209344">How many distinct ways that silence will occur?</a>
%H G. P. Michon, <a href="http://www.numericana.com/answer/counting.htm#scream">Brocoum's Screaming Circles</a>.
%H G. P. Michon, <a href="http://www.numericana.com/answer/graphs.htm#alekseyev">Silent circles</a>, enumerated by Max Alekseyev.
%H G. P. Michon, <a href="http://www.numericana.com/answer/graphs.htm#prisms">A screaming game for short-sighted people</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16,10,-1).
%F a(n) = 8*a(n-1) - 16*a(n-2) + 10*a(n-3) - a(n-4), for n > 1.
%F O.g.f.: 2*x^2*(15 -42*x +29*x^2 -3*x^3)/((1-x)*(1-7*x+9*x^2-x^3)). - _R. J. Mathar_, Jun 16 2008
%F a(n) = -7*[n=1] + (A141385(n) - 1). - _G. C. Greubel_, Mar 31 2021
%e a(1)=0 because two people always make eye contact when they look at each other.
%e a(2)=30 because 4 people can look at each other in 30 distinct ways without making eye contact.
%t Join[{0}, LinearRecurrence[{8, -16, 10, -1}, {30, 156, 826, 4406}, 20]] (* _Jean-François Alcover_, Dec 14 2018 *)
%o (Magma) I:=[30, 156, 826, 4406]; [0] cat [n le 4 select I[n] else 8*Self(n-1) -16*Self(n-2) +10*Self(n-3) -Self(n-4): n in [1..30]]; // _G. C. Greubel_, Mar 31 2021
%o (Sage)
%o def A141221_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( 2*x^2*(15 -42*x +29*x^2 -3*x^3)/((1-x)*(1-7*x+9*x^2-x^3)) ).list()
%o a=A141221_list(30); a[1;] # _G. C. Greubel_, Mar 31 2021
%Y Cf. A094047, A114939.
%Y Cf. A141384, A141385.
%K nonn
%O 1,2
%A _Max Alekseyev_, Jun 14 2008
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