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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. 12
 1, 3, 13, 59, 269, 1227, 5597, 25531, 116461, 531243, 2423293, 11053979, 50423309, 230008587, 1049196317, 4785964411, 21831429421, 99585218283, 454263232573, 2072145726299, 9452202166349, 43116719379147, 196679192563037 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) = A020698(n) - 4*A020698(n-1) + 4*A020698(n-2) (n>=2). Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005 REFERENCES S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78). Stanley, Richard P. "Some Linear Recurrences Motivated by Stern’s Diatomic Array." The American Mathematical Monthly 127.2 (2020): 99-111. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1058 Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019. See p. 3. Zeying Xu, Graphical zonotopes with the same face vector, arXiv:1809.08764 [math.CO], 2018. Index entries for linear recurrences with constant coefficients, signature (5,-2). FORMULA a(n) = A005824(2n). G.f.: (1-2*x)/(1-5*x+2*x^2). a(n) = Sum_{alpha=RootOf(1-5*z+2*z^2)} (1 + 6*alpha)*alpha^(-1-n)/17. 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 a(n-1) = Sum_{k=0..n} A147703(n,k)*(-1)^k*2^(n-k), n>1. - Philippe Deléham, Nov 29 2008 a(n) = (a(n-1)^2 + 2^n)/a(n-2). - Irene Sermon, Oct 29 2013 a(n) = A107839(n) - 2*A107839(n-1). - R. J. Mathar, Feb 27 2019 MAPLE spec:= [S, {S=Sequence(Union(Prod(Sequence(Union(Z, Z)), Union(Z, Z)), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20); 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 MATHEMATICA a[0]=1; a[1]=3; a[n_]:= a[n] = 5a[n-1]-2a[n-2]; Table[ a[n], {n, 0, 30}] LinearRecurrence[{5, -2}, {1, 3}, 30] (* Harvey P. Dale, Apr 08 2014 *) CoefficientList[Series[(1-2x)/(1-5x+2x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 09 2014 *) PROG (PARI) Vec((1-2*x)/(1-5*x+2*x^2)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2011 (Magma) R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x)/(1-5*x+2*x^2) )); // G. C. Greubel, Feb 10 2019 (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 (Sage) def A052984_list(prec):     P. = PowerSeriesRing(ZZ, prec)     return P( (1-2*x)/(1-5*x+2*x^2) ).list() A052984_list(30) # G. C. Greubel, Feb 10 2019 (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 CROSSREFS Cf. A005824, A020698. Sequence in context: A268596 A199297 A152594 * A262664 A151229 A333472 Adjacent sequences:  A052981 A052982 A052983 * A052985 A052986 A052987 KEYWORD nonn,easy AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 EXTENSIONS Edited by Robert G. Wilson v, Dec 29 2002 STATUS approved

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Last modified November 26 05:48 EST 2022. Contains 358353 sequences. (Running on oeis4.)