OFFSET
0,3
COMMENTS
a(k) is Heuberger and Wagner's G_k at lemma 6.2 (2). They show (theorem 3.3 (1)) that the largest number of maximum matchings in a tree of 7k+1 vertices is a(k+1) and there is a unique free tree with this many maximum matchings. (See A333347 for all tree sizes.) - Kevin Ryde, Apr 11 2020
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Clemens Heuberger and Stephan Wagner, The Number of Maximum Matchings in a Tree, Discrete Mathematics, volume 311, issue 21, November 2011, pages 2512-2542; arXiv preprint, arXiv:1011.6554 [math.CO], 2010.
Index entries for linear recurrences with constant coefficients, signature (11,-9).
FORMULA
a(n) = ((11+sqrt(85))^n-(11-sqrt(85))^n)/(2^n*sqrt(85)).
G.f.: x/(1-11*x+9*x^2). - Philippe Deléham, Feb 12 2012
E.g.f.: (2/sqrt(85))*exp(11*x/2)*sinh(sqrt(85)*x/2). - G. C. Greubel, Dec 18 2015
a(n) = (L^n - H^n)/(L-H) where L = (11+sqrt(85))/2 and H = (11-sqrt(85))/2. [Heuberger and Wagner lemma 6.2 (2)] - Kevin Ryde, Apr 11 2020
MATHEMATICA
LinearRecurrence[{11, -9}, {0, 1}, 50] (* T. D. Noe, May 23 2011 *)
PROG
(PARI) concat(0, Vec(x/(1-11*x+9*x^2) + O(x^100))) \\ Altug Alkan, Dec 18 2015
(PARI) a(n) = polcoeff(lift(Mod('x, 'x^2-11*'x+9)^n), 1); \\ Kevin Ryde, Apr 11 2020
(Magma) I:=[0, 1]; [n le 2 select I[n] else 11*Self(n-1)-9*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 19 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Rolf Pleisch, May 22 2011
EXTENSIONS
Extended by T. D. Noe, May 23 2011
STATUS
approved