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
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