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A052984 a(n) = 5*a(n-1) - 2*a(n-2) for n>1, with a(0) = 1, a(1) = 3.

%I A052984 #81 Jun 18 2025 16:07:23
%S A052984 1,3,13,59,269,1227,5597,25531,116461,531243,2423293,11053979,
%T A052984 50423309,230008587,1049196317,4785964411,21831429421,99585218283,
%U A052984 454263232573,2072145726299,9452202166349,43116719379147,196679192563037,897162524056891,4092454235158381
%N A052984 a(n) = 5*a(n-1) - 2*a(n-2) for n>1, with a(0) = 1, a(1) = 3.
%C A052984 a(n) = A020698(n) - 4*A020698(n-1) + 4*A020698(n-2) (n>=2). Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Jun 12 2005
%D A052984 S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).
%D A052984 Stanley, Richard P. "Some Linear Recurrences Motivated by Stern’s Diatomic Array." The American Mathematical Monthly 127.2 (2020): 99-111.
%H A052984 Vincenzo Librandi, <a href="/A052984/b052984.txt">Table of n, a(n) for n = 0..200</a>
%H A052984 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1058">Encyclopedia of Combinatorial Structures 1058</a>
%H A052984 Richard P. Stanley, <a href="https://arxiv.org/abs/1901.04647">Some Linear Recurrences Motivated by Stern's Diatomic Array</a>, arXiv:1901.04647 [math.CO], 2019. See p. 3.
%H A052984 Zeying Xu, <a href="https://arxiv.org/abs/1809.08764">Graphical zonotopes with the same face vector</a>, arXiv:1809.08764 [math.CO], 2018.
%H A052984 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-2).
%F A052984 a(n) = A005824(2n).
%F A052984 G.f.: (1-2*x)/(1-5*x+2*x^2).
%F A052984 a(n) = Sum_{alpha=RootOf(1-5*z+2*z^2)} (1 + 6*alpha)*alpha^(-1-n)/17.
%F A052984 a(n) = [M^(n+1)]_2,2, where M is the 3 X 3 matrix defined as follows: M = [2,1,2; 1,1,1; 2,1,2]. - _Simone Severini_, Jun 12 2006
%F A052984 a(n-1) = Sum_{k=0..n} A147703(n,k)*(-1)^k*2^(n-k), n>1. - _Philippe Deléham_, Nov 29 2008
%F A052984 a(n) = (a(n-1)^2 + 2^n)/a(n-2). - _Irene Sermon_, Oct 29 2013
%F A052984 a(n) = A107839(n) - 2*A107839(n-1). - _R. J. Mathar_, Feb 27 2019
%F A052984 E.g.f.: exp(5*x/2)*(sqrt(17)*cosh(sqrt(17)*x/2) + sinh(sqrt(17)*x/2))/sqrt(17). - _Stefano Spezia_, Jun 17 2025
%p A052984 spec:= [S,{S=Sequence(Union(Prod(Sequence(Union(Z,Z)),Union(Z,Z)),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%p A052984 a[0]:=1: a[1]:=3: for n from 2 to 25 do a[n]:=5*a[n-1]-2*a[n-2] od: seq(a[n],n=0..25); # _Emeric Deutsch_
%t A052984 a[0]=1; a[1]=3; a[n_]:= a[n] = 5a[n-1]-2a[n-2]; Table[ a[n], {n, 0, 30}]
%t A052984 LinearRecurrence[{5,-2},{1,3},30] (* _Harvey P. Dale_, Apr 08 2014 *)
%t A052984 CoefficientList[Series[(1-2x)/(1-5x+2x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Apr 09 2014 *)
%o A052984 (PARI) Vec((1-2*x)/(1-5*x+2*x^2)+O(x^30)) \\ _Charles R Greathouse IV_, Nov 20 2011
%o A052984 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x)/(1-5*x+2*x^2) )); // _G. C. Greubel_, Feb 10 2019
%o A052984 (Magma) a:=[1,3]; [n le 2 select a[n] else 5*Self(n-1)-2*Self(n-2):n in [1..25]]; // _Marius A. Burtea_, Oct 23 2019
%o A052984 (Sage)
%o A052984 def A052984_list(prec):
%o A052984     P.<x> = PowerSeriesRing(ZZ, prec)
%o A052984     return P( (1-2*x)/(1-5*x+2*x^2) ).list()
%o A052984 A052984_list(30) # _G. C. Greubel_, Feb 10 2019
%o A052984 (GAP) a:=[1,3];; for n in [3..30] do a[n]:=5*a[n-1]-2*a[n-2]; od; a; # _G. C. Greubel_, Oct 23 2019
%Y A052984 Cf. A005824, A020698, A107839, A147703.
%K A052984 nonn,easy
%O A052984 0,2
%A A052984 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052984 Edited by _Robert G. Wilson v_, Dec 29 2002