A106435 a(n) = 3*a(n-1) + 3*a(n-2), a(0)=0, a(1)=3.

0, 3, 9, 36, 135, 513, 1944, 7371, 27945, 105948, 401679, 1522881, 5773680, 21889683, 82990089, 314639316, 1192888215, 4522582593, 17146412424, 65006985051, 246460192425, 934401532428, 3542585174559, 13430960120961

Offset

0,2

Comments

The first entry of the vector v[n] = M*v[n-1], where M is the 2 x 2 matrix [[0,3],[1,3]] and v[1] is the column vector [0,1]. The characteristic polynomial of the matrix M is x^2-3x-3.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

Tanya Khovanova, Recursive Sequences

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

Formula

G.f.: 3*x/(1-3*x-3*x^2). - Philippe Deléham, Nov 19 2008

From G. C. Greubel, Mar 12 2020: (Start)

a(n) = 3^((n+1)/2) * Fibonacci(n, sqrt(3)), where F(n, x) is the Fibonacci polynomial.

a(n) = 3^((n+1)/2)*i^(1-n)*ChebyshevU(n-1, i*sqrt(3)/2). (End)

Maple

seq(coeff(series(3*x/(1-3*x-3*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Mar 12 2020

Mathematica

LinearRecurrence[{3, 3}, {0, 3}, 30] (* G. C. Greubel, Mar 12 2020 *)

Prog

(PARI) a(n)=([0, 3; 1, 3]^n)[1, 2]
(Haskell)
a106435 n = a106435_list !! n
a106435_list = 0 : 3 : map (* 3) (zipWith (+) a106435_list (tail
a106435_list))
-- Reinhard Zumkeller, Oct 15 2011
(Magma) a:=[0, 3]; [n le 2 select a[n] else    3*Self(n-1) + 3*Self(n-2) : n in [1..24]]; // Marius A. Burtea, Jan 21 2020
(Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!(3*x/(1-3*x-3*x^2))); // Marius A. Burtea, Jan 21 2020
(Sage) [3^((n+1)/2)*i^(1-n)*chebyshev_U(n-1, i*sqrt(3)/2) for n in (0..30)] # G. C. Greubel, Mar 12 2020

Crossrefs

Equals 3*A030195(n).

Cf. A028860.

Cf. A002605, A026150, A028859, A080040, A083337, A108898, A125145.

Sequence in context: A057390 A183495 A185162 * A276368 A058540 A350451

Adjacent sequences: A106432 A106433 A106434 * A106436 A106437 A106438

Keyword

nonn,easy

Author

Roger L. Bagula, May 29 2005

Extensions

Edited by N. J. A. Sloane, May 20 2006 and May 29 2006

Offset corrected by Reinhard Zumkeller, Oct 15 2011

Status

approved