OFFSET
1,2
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).
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