A190872 a(n) = 11*a(n-1) - 9*a(n-2), a(0)=0, a(1)=1.

0, 1, 11, 112, 1133, 11455, 115808, 1170793, 11836451, 119663824, 1209774005, 12230539639, 123647969984, 1250052813073, 12637749213947, 127764766035760, 1291672683467837, 13058516623824367, 132018628710857504, 1334678266205013241, 13493293269857428115

Offset: 0

Author

Rolf Pleisch, May 22 2011

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A333345 (growth power), A190871, A190873.

Programs

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

Mathematica

LinearRecurrence[{11, -9}, {0, 1}, 50] (* T. D. Noe, May 23 2011 *)

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

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

Extensions

Extended by T. D. Noe, May 23 2011