A051927 Number of independent vertex sets in the n-prism graph Y_n = K_2 X C_n (n > 2).

3, 1, 7, 13, 35, 81, 199, 477, 1155, 2785, 6727, 16237, 39203, 94641, 228487, 551613, 1331715, 3215041, 7761799, 18738637, 45239075, 109216785, 263672647, 636562077, 1536796803, 3710155681, 8957108167, 21624372013, 52205852195

Offset: 0

Author

Stephen G Penrice, Dec 19 1999

Keywords

• easy
• nonn

Comments

For n>1, a(n) is also the number of ways to place k non-attacking wazirs on a 2 X n horizontal cylinder, summed over all k>=0 (wazir is a leaper [0,1]). - Vaclav Kotesovec, May 08 2012

Also the number of vertex covers for Y_n. - Eric W. Weisstein, Jan 04 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 400-401.
• Eric Weisstein's World of Mathematics, Independent Vertex Set
• Eric Weisstein's World of Mathematics, Prism Graph
• Eric Weisstein's World of Mathematics, Vertex Cover
• Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Crossrefs

Column 2 of A286513 and row 2 of A287376.

Cf. A000129, A002203.

Programs

Magma

I:=[3, 1, 7]; [n le 3 select I[n] else Self(n-1) + 3*Self(n-2) + Self(n-3): n in [1..30]]; // Vincenzo Librandi, May 04 2013

Maple

A051927 := x -> (1+sqrt(2))^x+(-1)^x+(1-sqrt(2))^x;
seq(simplify(A051927(i)),i=0..28); # Peter Luschny, Aug 13 2012

Mathematica

CoefficientList[Series[(3 - 2 x - 3 x^2) / ((1 - 2 x - x^2) (1 + x)), {x, 0, 40}], x] (* Vincenzo Librandi, May 04 2013 *)
Table[LucasL[n, 2] + (-1)^n, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
LinearRecurrence[{1, 3, 1}, {1, 7, 13}, {0, 20}] (* Eric W. Weisstein, Sep 27 2017 *)

PARI

a(n)=polcoeff((3-2*x-3*x^2)/(1-2*x-x^2)/(1+x)+x*O(x^n),n)

PARI

x='x+O('x^66); Vec( (3-2*x-3*x^2)/((1-2*x-x^2)*(1+x)) ) \\ Joerg Arndt, May 04 2013

Sage

def A051927(x) : return (1+sqrt(2))^x+(-1)^x+(1-sqrt(2))^x
[A051927(i).round() for i in (0..28)] # Peter Luschny, Aug 13 2012

Formula

a(n) = a(n-1) + 3*a(n-2) + a(n-3).

G.f.: (3-2x-3x^2)/((1-2x-x^2)(1+x)). - Michael Somos, Apr 07 2003

Let A=[0, 1, 1;1, 1, 1;1, 1, 0] be the adjacency matrix of a triangle with a loop at a vertex. Then a(n)=trace(A^n). a(n)=(-1)^n+(1-sqrt(2))^n+(1+sqrt(2))^n. - Paul Barry, Jul 22 2004

a(n) = A002203(n) + (-1)^n. - Vladimir Reshetnikov, Sep 15 2016

a(n)+a(n+1) = 4*A000129(n+1). - R. J. Mathar, Feb 13 2020

E.g.f.: cosh(x) + 2*exp(x)*cosh(sqrt(2)*x) - sinh(x). - Stefano Spezia, Mar 31 2024