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 A051927 Number of independent vertex sets in the n-prism graph Y_n = K_2 X C_n (n > 2). 14
 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 (list; graph; refs; listen; history; text; internal format)
 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 Column 2 of A286513 and row 2 of A287376. Cf. A000129, A002203. Sequence in context: A370381 A161380 A257852 * A322069 A194595 A219063 Adjacent sequences: A051924 A051925 A051926 * A051928 A051929 A051930 KEYWORD easy,nonn AUTHOR Stephen G Penrice, Dec 19 1999 STATUS approved

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Last modified July 12 19:17 EDT 2024. Contains 374252 sequences. (Running on oeis4.)