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A108300 a(n+2) = 3*a(n+1) + a(n), with a(0) = 1, a(1) = 5.

%I A108300 #58 Jul 26 2024 15:11:13
%S A108300 1,5,16,53,175,578,1909,6305,20824,68777,227155,750242,2477881,
%T A108300 8183885,27029536,89272493,294847015,973813538,3216287629,10622676425,
%U A108300 35084316904,115875627137,382711198315,1264009222082,4174738864561,13788225815765,45539416311856
%N A108300 a(n+2) = 3*a(n+1) + a(n), with a(0) = 1, a(1) = 5.
%C A108300 Binomial transform is A109114.
%C A108300 Invert transform is A109115.
%C A108300 Inverse invert transform is A016777.
%C A108300 Inverse binomial transform is A006130.
%H A108300 Andrew Howroyd, <a href="/A108300/b108300.txt">Table of n, a(n) for n = 0..1000</a>
%H A108300 Sergio Falcon, <a href="https://doi.org/10.1016/j.chaos.2016.03.038">The k-Fibonacci difference sequences</a>, Chaos, Solitons & Fractals, Volume 87, June 2016, Pages 153-157.
%H A108300 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A108300 Vincent Vatter, <a href="https://arxiv.org/abs/1605.04297">Growth rates of permutation classes: from countable to uncountable</a>, arXiv:1605.04297 [math.CO], 2016. (Mentions a signed version.)
%H A108300 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,1).
%F A108300 G.f.: (1 + 2*x)/(1 - 3*x - x^2).
%F A108300 a(n) = A052924(n+1) - A052924(n).
%F A108300 a(n)*a(n-2) = a(n-1)^2 + 9*(-1)^n. - _Roger L. Bagula_, May 17 2010
%F A108300 a(n) = 3^n*Sum_{k=0..n} A374439(n, k)*(1/3)^k. - _Peter Luschny_, Jul 26 2024
%p A108300 seriestolist(series((-2*x-1)/(x^2-1+3*x), x=0,25));
%t A108300 LinearRecurrence[{3,1},{1,5},40] (* _Harvey P. Dale_, Jul 04 2013 *)
%o A108300 (PARI) Vec((1 + 2*x)/(1 - 3*x - x^2) + O(x^30)) \\ _Andrew Howroyd_, Jun 05 2021
%Y A108300 Row sums and main diagonal of A143972. - _Gary W. Adamson_, Sep 06 2008
%Y A108300 Cf. A109114, A109115, A016777, A006130, A000004, A052924, A228916, A374439.
%K A108300 nonn,easy
%O A108300 0,2
%A A108300 _Creighton Dement_, Jul 24 2005