A141384 -id:A141384 - cp's OEIS frontend

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

Max Alekseyev, Jun 14 2008

nonn

			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.

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).

Cf. A094047, A114939.

Cf. A141384, A141385.

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

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

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

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

A141385 a(n) = 7a(n-1) - 9a(n-2) + a(n-3) with a(0)=3, a(1)=7, a(2)=31.

Original entry on oeis.org

3, 7, 31, 157, 827, 4407, 23563, 126105, 675075, 3614143, 19349431, 103593805, 554625899, 2969386479, 15897666067, 85113810057, 455687062275, 2439682811479, 13061709929935, 69930511268509, 374397872321627

Offset: 0

Gerard P. Michon, Jul 02 2008, Jul 23 2008

The old definition given for this sequence was "A sequence obeying a third-order linear recurrence".
Ruling out finitely many exceptional terms, this sequence differs by a constant from several related enumerations with a slightly more complicated structure (fourth-order linear recurrence):
For n>0, A141221(n) = a(n) - 1. For n>2, A141384(n) = a(n) + 1.

			a(0) = 3 = A^0 + B^0 + C^0, a(1) = 7 = A + B + C.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. P. Michon, Silent Prisms: A Screaming Game for Short-Sighted People.
Index entries for linear recurrences with constant coefficients, signature (7,-9,1).

Cf. A141221, A141384.

I:=[3,7,31]; [n le 3 select I[n] else 7*Self(n-1)-9*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 21 2012

m:=30; S:=series( (3-14*x+9*x^2)/(1-7*x+9*x^2-x^3), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 30 2021

LinearRecurrence[{7,-9,1},{3,7,31},40] (* Harvey P. Dale, May 25 2011 *)
CoefficientList[Series[(3 -14x +9x^2)/(1 -7x +9x^2 -x^3), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 21 2012 *)

a(n)=([0,1,0; 0,0,1; 1,-9,7]^n*[3;7;31])[1,1] \\ Charles R Greathouse IV, Feb 10 2017

def A141385_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (3-14*x+9*x^2)/(1-7*x+9*x^2-x^3) ).list()
A141385_list(40) # G. C. Greubel, Mar 30 2021

G.f.: (3 - 14*x + 9*x^2)/(1 - 7*x + 9*x^2 - x^3).
a(n+3) = 7*a(n+2) - 9*a(n+1) + a(n).
a(n) = A^n + B^n + C^n, where, putting u = atan(sqrt(5319)/73), we have:
A = 5.3538557854308282... = (7 + 2*sqrt(22)*cos(u/3))/3,
B = 1.5235479602692093... = (7 - sqrt(22)*cos(u/3) + sqrt(66)*sin(u/3))/3,
C = 0.1225962542999624... = (7 - sqrt(22)*cos(u/3) - sqrt(66)*sin(u/3))/3.

New definition by Bruno Berselli, Oct 22 2012

cp's OEIS Frontend

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.

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A141385 a(n) = 7a(n-1) - 9a(n-2) + a(n-3) with a(0)=3, a(1)=7, a(2)=31.

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cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A141385 a(n) = 7*a(n-1) - 9*a(n-2) + a(n-3) with a(0)=3, a(1)=7, a(2)=31.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A141385 a(n) = 7a(n-1) - 9a(n-2) + a(n-3) with a(0)=3, a(1)=7, a(2)=31.