%I #93 Jan 15 2026 15:23:41

%S 1,3,13,59,269,1227,5597,25531,116461,531243,2423293,11053979,

%T 50423309,230008587,1049196317,4785964411,21831429421,99585218283,

%U 454263232573,2072145726299,9452202166349,43116719379147,196679192563037,897162524056891,4092454235158381

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

%C a(n) = A020698(n) - 4*A020698(n-1) + 4*A020698(n-2) (n>=2). Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Jun 12 2005

%C a(n) is also the sum of the squares of the coefficients of Product_{k=0..n-1} 1 + x^2^k + x^2^(k+1), cf. formula. If both exponents are incremented by 1, the same procedure gives A385082, where we know no formula or recurrence relation. - _M. F. Hasler_, Jan 08 2026

%D 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 Stanley, Richard P. "Some Linear Recurrences Motivated by Stern’s Diatomic Array." The American Mathematical Monthly 127.2 (2020): 99-111.

%H Vincenzo Librandi, <a href="/A052984/b052984.txt">Table of n, a(n) for n = 0..200</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1058">Encyclopedia of Combinatorial Structures 1058</a>

%H 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 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-2).

%F a(n) = A005824(2n).

%F G.f.: (1-2*x)/(1-5*x+2*x^2).

%F a(n) = Sum_{alpha=RootOf(1-5*z+2*z^2)} (1 + 6*alpha)*alpha^(-1-n)/17.

%F 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 a(n-1) = Sum_{k=0..n} A147703(n,k)*(-1)^k*2^(n-k), n>1. - _Philippe Deléham_, Nov 29 2008

%F a(n) = (a(n-1)^2 + 2^n)/a(n-2). - _Irene Sermon_, Oct 29 2013

%F a(n) = A107839(n) - 2*A107839(n-1). - _R. J. Mathar_, Feb 27 2019

%F 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

%F a(n) = Sum_{k>=0} b(n,k)^2 where Sum_{k>=0} b(n,k)*x^k = Product_{k=0..n-1} 1 + x^2^k + x^2^(k+1), see also A385082. - _M. F. Hasler_, Jan 08 2026

%p 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 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_, Jun 12 2005

%t a[0]=1; a[1]=3; a[n_]:= a[n] = 5a[n-1]-2a[n-2]; Table[ a[n], {n, 0, 30}]

%t LinearRecurrence[{5,-2},{1,3},30] (* _Harvey P. Dale_, Apr 08 2014 *)

%t CoefficientList[Series[(1-2x)/(1-5x+2x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Apr 09 2014 *)

%o (PARI) Vec((1-2*x)/(1-5*x+2*x^2)+O(x^30)) \\ _Charles R Greathouse IV_, Nov 20 2011

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x)/(1-5*x+2*x^2) )); // _G. C. Greubel_, Feb 10 2019

%o (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 (SageMath)

%o def A052984_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (1-2*x)/(1-5*x+2*x^2) ).list()

%o A052984_list(30) # _G. C. Greubel_, Feb 10 2019

%o (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 Cf. A005824 (of which this is a bisection), A020698, A107839, A147703, A385082 (cf. comments).

%K nonn,easy

%O 0,2

%A _INRIA Encyclopedia of Combinatorial Structures_, Jan 25 2000

%E Edited by _Robert G. Wilson v_, Dec 29 2002