OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Yuhan Jiang, The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model, arXiv:2312.09427 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (6,-4).
FORMULA
a(n) = Sum_{k=0..2*n} Trinomial(n,k)*Lucas(k+1), where Trinomial(n,k) = trinomial coefficients (A027907).
a(n) = 2^n*Lucas(2*n+1), where Lucas = A000032.
From Philippe Deléham, Mar 01 2004: (Start)
a(0) = 1, a(1) = 8, a(n+1) = 6*a(n) - 4*a(n-1). (End)
From Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009: (Start)
a(n) = ((1+sqrt(5))*(3+sqrt(5))^n + (1-sqrt(5))*(3-sqrt(5))^n)/2.
Third binomial transform of 1, 5, 5, 25, 25, 125. (End)
G.f.: (1 + 2*x)/(1 - 6*x + 4*x^2). - Colin Barker, Mar 23 2012
MATHEMATICA
a[n_]:=(MatrixPower[{{2, 2}, {2, 4}}, n].{{2}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(3 + s)^n + (1 - s)(3 - s)^n)/2]]; Array[f, 21, 0] (* Robert G. Wilson v, Mar 07 2011 *)
LinearRecurrence[{6, -4}, {1, 8}, 30] (* G. C. Greubel, Dec 21 2017 *)
PROG
(PARI) x='x+O('x^30); Vec((1 + 2*x)/(1 - 6*x + 4*x^2)) \\ G. C. Greubel, Dec 21 2017
(Magma) I:=[1, 8]; [n le 2 select I[n] else 6*Self(n-1)-4*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 21 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emanuele Munarini, May 21 2003
STATUS
approved