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A180148
a(n) = 3*a(n-1) + a(n-2) with a(0)=2 and a(1)=5.
5
2, 5, 17, 56, 185, 611, 2018, 6665, 22013, 72704, 240125, 793079, 2619362, 8651165, 28572857, 94369736, 311682065, 1029415931, 3399929858, 11229205505, 37087546373, 122491844624, 404563080245, 1336181085359, 4413106336322, 14575500094325, 48139606619297
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
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
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.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^n*2/(A006497(n) - A006190(n)*sqrt(13)).
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
a(n) = 2*A006190(n+1) - A006190(n). - 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
Appears in A180142.
Cf. A000602 (more information on n-alkanes).
Sequence in context: A121193 A159866 A042671 * A241133 A148410 A190531
KEYWORD
easy,nonn
AUTHOR
Johannes W. Meijer, Aug 13 2010
STATUS
approved