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A190872
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a(n) = 11*a(n-1) - 9*a(n-2), a(0)=0, a(1)=1.
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11
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0, 1, 11, 112, 1133, 11455, 115808, 1170793, 11836451, 119663824, 1209774005, 12230539639, 123647969984, 1250052813073, 12637749213947, 127764766035760, 1291672683467837, 13058516623824367, 132018628710857504, 1334678266205013241, 13493293269857428115
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = ((11+sqrt(85))^n-(11-sqrt(85))^n)/(2^n*sqrt(85)).
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
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MATHEMATICA
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LinearRecurrence[{11, -9}, {0, 1}, 50] (* T. D. Noe, May 23 2011 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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