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.

0, 30, 156, 826, 4406, 23562, 126104, 675074, 3614142, 19349430, 103593804, 554625898, 2969386478, 15897666066, 85113810056, 455687062274, 2439682811478, 13061709929934, 69930511268508, 374397872321626

Offset: 1

Author

Max Alekseyev, Jun 14 2008

Keywords

• nonn

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Max A. Alekseyev and Gérard P. Michon, Making Walks Count: From Silent Circles to Hamiltonian Cycles, arXiv:1602.01396 [math.CO], 2016.
• Art of Problem Solving Forum, How many distinct ways that silence will occur?
• G. P. Michon, Brocoum's Screaming Circles.
• G. P. Michon, Silent circles enumerated by Max Alekseyev.
• G. P. Michon, A screaming game for short-sighted people.
• Index entries for linear recurrences with constant coefficients, signature (8,-16,10,-1).

Crossrefs

Cf. A094047, A114939.

Cf. A141384, A141385.

Programs

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

Mathematica

Join[{0}, LinearRecurrence[{8, -16, 10, -1}, {30, 156, 826, 4406}, 20]] (* Jean-François Alcover, Dec 14 2018 *)

Sage

def A141221_list(prec):
    P. = 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)
print(a[1:]) # G. C. Greubel, Mar 31 2021

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