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
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
a(n) is the number of subgraphs of the path graph on n vertices with a loop attached to each vertex. Compare to A001519 which is the same but without attaching loops. - Nick Krempel, Apr 21 2026
From Greg Dresden and Xinyan Lu, Jun 19 2026: (Start)
a(n) is the number of ways to tile this fang sheng type of pattern (of length n with 3n rhombus-shaped cells in total) using only single and double tiles (made up of a single rhombus, or two connected rhombi). Shown here is the template for n=4, and there are a(4) = 269 possible tilings.
/\ /\ /\ /\
/ \/ \/ \/ \
/\ /\ /\ /\ /
/ \/ \/ \/ \/
\ /\ /\ /\ /\
\/ \/ \/ \/ \
\ /\ /\ /\ /
\/ \/ \/ \/
We give one of them as an example, using four double-tiles and four single-tiles. .
/\ /\ /\ /\
/ \/ \/ \/ \
/ /\ / /\ /
/ / \/ / \/
\ / /\ /\ \
\/ / \/ \ \
\ /\ /\ /\ /
\/ \/ \/ \/ . (End)
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
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
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
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, Jun 12 2005
MATHEMATICA
a[0]=1; a[1]=3; a[n_]:= a[n] = 5a[n-1]-2a[n-2]; Table[ a[n], {n, 0, 30}]
(* Alternative: *)
LinearRecurrence[{5, -2}, {1, 3}, 30] (* Harvey P. Dale, Apr 08 2014 *)
(* Alternative: *)
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<x>:=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
(SageMath)
def A052984_list(prec):
P.<x> = 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
KEYWORD
nonn,easy
AUTHOR
INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000
EXTENSIONS
Edited by Robert G. Wilson v, Dec 29 2002
STATUS
approved