A141015 a(0) = 0, a(1) = 1, a(2) = 2; for n > 2, a(n) = a(n-1) + 2*a(n-2) + a(n-3).

0, 1, 2, 4, 9, 19, 41, 88, 189, 406, 872, 1873, 4023, 8641, 18560, 39865, 85626, 183916, 395033, 848491, 1822473, 3914488, 8407925, 18059374, 38789712, 83316385, 178955183, 384377665, 825604416, 1773314929, 3808901426

Offset

0,3

Comments

Central axis of triangle G(n, k): G(n,0) = G(n+1, n+1) = 1, G(n+2, n+1) = 2, G(n+3, n+1) = 4, G(n+4, k) = G(n+1, k-1) + G(n+1, k) + G(n+2, k) + G(n+3, k) for k = 1..(n+1). (This is triangular array A140997.)

Central axis of triangle G(n, k): G(n, n) = G(n+1, 0) = 1, G(n+2, 1) = 2, G(n+3, 2) = 4, G(n+4, k) = G(n+1, k-2) + G(n+1, k-3) + G(n+2, k-2) + G(n+3, k-1) for k = 3..(n+3). (This is triangular array A140994, which is a mirror image of A140997.)

a(n-1) is the top left entry of the n-th power of any of the 3X3 matrices [0, 1, 1; 1, 1, 1; 0, 1, 0], [0, 1, 0; 1, 1, 1; 1, 1, 0], [0, 1, 1; 0, 0, 1; 1, 1, 1] or [0, 0, 1; 1, 0, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,2,1).

Formula

From R. J. Mathar, Aug 22 2008: (Start)

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

a(n) = (-1)^(n+1)*A078039(n-1). (End)

Mathematica

CoefficientList[Series[x (1 + x)/(1 - x - 2 x^2 - x^3), {x, 0, 50}], x] (* G. C. Greubel, Jun 09 2017 *)

Prog

(Sage) from sage.combinat.sloane_functions import recur_gen3; it = recur_gen3(0, 1, 2, 1, 2, 1); [next(it) for i in range(31)] # Zerinvary Lajos, May 17 2009
(PARI) x='x+O('x^50); concat([0], Vec(x*(1+x)/(1-x-2*x^2-x^3))) \\ G. C. Greubel, Jun 09 2017

Crossrefs

Cf. A000129, A140994, A140997.

Sequence in context: A184936 A330489 A299106 * A141683 A142474 A078039

Adjacent sequences: A141012 A141013 A141014 * A141016 A141017 A141018

Keyword

nonn,easy

Author

Juri-Stepan Gerasimov, Jul 11 2008

Extensions

Typo in definition corrected by Paolo P. Lava, Jul 31 2008

Dysfunctional Maple program removed by R. J. Mathar, Oct 28 2009

Comments clarified by Petros Hadjicostas, Jun 12 2019

Status

approved