A108300 a(n) = 3*a(n-1) + a(n-2), 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.

a(n) is the total number of vertex clique partitions of the triangular snake graph TS_n, where n is the number of triangles in the graph. a(3)=53 is the number of vertex clique partitions of TS_3

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.___.___.___. - Birhanu Gebrehanna Habtemariam, May 19 2026

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

E.g.f.: exp(3*x/2)*(13*cosh(sqrt(13)*x/2) + 7*sqrt(13)*sinh(sqrt(13)*x/2))/13. - Stefano Spezia, May 27 2026

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