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 A190872 a(n) = 11*a(n-1) - 9*a(n-2), a(0)=0, a(1)=1. 10

%I

%S 0,1,11,112,1133,11455,115808,1170793,11836451,119663824,1209774005,

%T 12230539639,123647969984,1250052813073,12637749213947,

%U 127764766035760,1291672683467837,13058516623824367,132018628710857504,1334678266205013241,13493293269857428115

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

%C 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

%H G. C. Greubel, <a href="/A190872/b190872.txt">Table of n, a(n) for n = 0..1000</a>

%H Clemens Heuberger and Stephan Wagner, <a href="https://doi.org/10.1016/j.disc.2011.07.028">The Number of Maximum Matchings in a Tree</a>, Discrete Mathematics, volume 311, issue 21, November 2011, pages 2512-2542; <a href="https://arxiv.org/abs/1011.6554">arXiv preprint</a>, arXiv:1011.6554 [math.CO], 2010.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-9).

%F a(n) = ((11+sqrt(85))^n-(11-sqrt(85))^n)/(2^n*sqrt(85)).

%F G.f.: x/(1-11*x+9*x^2). - _Philippe Deléham_, Feb 12 2012

%F E.g.f.: (2/sqrt(85))*exp(11*x/2)*sinh(sqrt(85)*x/2). - _G. C. Greubel_, Dec 18 2015

%F 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

%t LinearRecurrence[{11, -9}, {0, 1}, 50] (* _T. D. Noe_, May 23 2011 *)

%o (PARI) concat(0, Vec(x/(1-11*x+9*x^2) + O(x^100))) \\ _Altug Alkan_, Dec 18 2015

%o (PARI) a(n) = polcoeff(lift(Mod('x,'x^2-11*'x+9)^n), 1); \\ _Kevin Ryde_, Apr 11 2020

%o (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

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

%K nonn,easy

%O 0,3

%A _Rolf Pleisch_, May 22 2011

%E Extended by _T. D. Noe_, May 23 2011

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Last modified August 14 19:09 EDT 2020. Contains 336483 sequences. (Running on oeis4.)