A108300 a(n+2) = 3*a(n+1) + a(n), with a(0) = 1, a(1) = 5.

1, 5, 16, 53, 175, 578, 1909, 6305, 20824, 68777, 227155, 750242, 2477881, 8183885, 27029536, 89272493, 294847015, 973813538, 3216287629, 10622676425, 35084316904, 115875627137, 382711198315, 1264009222082, 4174738864561, 13788225815765, 45539416311856

Offset

0,2

Comments

Binomial transform is A109114.

Invert transform is A109115.

Inverse invert transform is A016777.

Inverse binomial transform is A006130.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Sergio Falcon, The k-Fibonacci difference sequences, Chaos, Solitons & Fractals, Volume 87, June 2016, Pages 153-157.

Tanya Khovanova, Recursive Sequences

Vincent Vatter, Growth rates of permutation classes: from countable to uncountable, arXiv:1605.04297 [math.CO], 2016. (Mentions a signed version.)

Index entries for linear recurrences with constant coefficients, signature (3,1).

Formula

G.f.: (1 + 2*x)/(1 - 3*x - x^2).

a(n) = A052924(n+1) - A052924(n).

a(n)*a(n-2) = a(n-1)^2 + 9*(-1)^n. - Roger L. Bagula, May 17 2010

a(n) = 3^n*Sum_{k=0..n} A374439(n, k)*(1/3)^k. - Peter Luschny, Jul 26 2024

Maple

seriestolist(series((-2*x-1)/(x^2-1+3*x), x=0, 25));

Mathematica

LinearRecurrence[{3, 1}, {1, 5}, 40] (* Harvey P. Dale, Jul 04 2013 *)

Prog

(PARI) Vec((1 + 2*x)/(1 - 3*x - x^2) + O(x^30)) \\ Andrew Howroyd, Jun 05 2021

Crossrefs

Row sums and main diagonal of A143972. - Gary W. Adamson, Sep 06 2008

Cf. A109114, A109115, A016777, A006130, A000004, A052924, A228916, A374439.

Sequence in context: A147536 A347434 A173871 * A041469 A089102 A098912

Adjacent sequences: A108297 A108298 A108299 * A108301 A108302 A108303

Keyword

nonn,easy

Author

Creighton Dement, Jul 24 2005

Status

approved