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A141221
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.
3
0, 30, 156, 826, 4406, 23562, 126104, 675074, 3614142, 19349430, 103593804, 554625898, 2969386478, 15897666066, 85113810056, 455687062274, 2439682811478, 13061709929934, 69930511268508, 374397872321626
OFFSET
1,2
LINKS
Max A. Alekseyev and Gérard P. Michon, Making Walks Count: From Silent Circles to Hamiltonian Cycles, arXiv:1602.01396 [math.CO], 2016.
G. P. Michon, Silent circles, enumerated by Max Alekseyev.
FORMULA
a(n) = 8*a(n-1) - 16*a(n-2) + 10*a(n-3) - a(n-4), for n > 1.
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
a(n) = -7*[n=1] + (A141385(n) - 1). - G. C. Greubel, Mar 31 2021
EXAMPLE
a(1)=0 because two people always make eye contact when they look at each other.
a(2)=30 because 4 people can look at each other in 30 distinct ways without making eye contact.
MATHEMATICA
Join[{0}, LinearRecurrence[{8, -16, 10, -1}, {30, 156, 826, 4406}, 20]] (* Jean-François Alcover, Dec 14 2018 *)
PROG
(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
(Sage)
def A141221_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
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()
a=A141221_list(30); a[1; ] # G. C. Greubel, Mar 31 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Max Alekseyev, Jun 14 2008
STATUS
approved