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

Author

Juri-Stepan Gerasimov, Jul 11 2008

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000129, A140994, A140997.

Programs

Mathematica

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

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

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

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)

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