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
(SageMath) 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
Dysfunctional Maple program removed by R. J. Mathar, Oct 28 2009
Comments clarified by Petros Hadjicostas, Jun 12 2019
STATUS
approved
