OFFSET
0,1
COMMENTS
Inverse binomial transform of A052961 (without the leading 1).
For n >= 1, also the number of matchings in the n-alkane graph. - Eric W. Weisstein, Jul 14 2021
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Alkane Graph
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set
Index entries for linear recurrences with constant coefficients, signature (3,1).
FORMULA
G.f.: (2-x)/(1-3*x-x^2).
a(n) = 3*a(n-1) + a(n-2) with a(0)=2 and a(1)=5.
a(n) = ((4+7*A)*A^(-n-1) + (4+7*B)*B^(-n-1))/13 with A = (-3+sqrt(13))/2 and B = (-3-sqrt(13))/2.
a(n) = 2 * Sum_{k=0..n-2} A168561(n-2,k)*3^k + 5 * Sum_{k=0..n-1} A168561(n-1,k)*3^k, n>0. - R. J. Mathar, Feb 14 2024
MAPLE
a:= n-> (<<0|1>, <1|3>>^n. <<2, 5>>)[1, 1]:
seq(a(n), n=0..27); # Alois P. Heinz, Jul 14 2021
MATHEMATICA
LinearRecurrence[{3, 1}, {5, 7}, 20] (* Eric W. Weisstein, Jul 14 2021 *)
CoefficientList[Series[(2 - x)/(1 - 3 x - x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jul 14 2021 *)
PROG
(PARI) a(n)=([0, 1; 1, 3]^n*[2; 5])[1, 1] \\ Charles R Greathouse IV, Oct 13 2016
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Johannes W. Meijer, Aug 13 2010
STATUS
approved