OFFSET
0,1
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).
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
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 *)
PROG
(PARI) a(n)=polcoeff((3-2*x-3*x^2)/(1-2*x-x^2)/(1+x)+x*O(x^n), n)
(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
(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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Stephen G Penrice, Dec 19 1999
STATUS
approved