OFFSET
1,2
COMMENTS
A102000 is generated from a 4 X 4 matrix, same format. A102002 is another recursive (1,2,4) sequence, generated from the matrix [0 1 0 / 0 0 1 / 4 2 1]. a(n)/a(n-1) tends to 2.46750385... an eigenvalue of M and a root of the characteristic polynomial x^3 - x^2 - 2x - 4.
With offset=0, a(n) is the number of length n sequences on alphabet {0,1,2} such that every set of three consecutive elements contains at least one 2. - Geoffrey Critzer, Feb 01 2012
Number of words of length n over the alphabet {1,2,3} such that no three odd letters appear consecutively. - Armend Shabani, Feb 28 2017
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,4).
FORMULA
a(n) = a(n-1) + 2*a(n-2) + 4*a(n-3), n>3. a(n) = left term in M^n * [1 0 0], where M = the 3 X 3 matrix [1 1 1 / 2 0 0 / 0 2 0].
a(n) = Sum{k=0..n} T(n-k, k)2^k, T(n, k) = trinomial coefficients (A027907). - Paul Barry, Feb 15 2005
G.f.: x*(1+2*x+4*x^2) / (1-x-2*x^2-4*x^3). - Geoffrey Critzer, Feb 01 2012, corrected by Armend Shabani, Feb 28 2017
G.f.: 1/(1-x-2*x^2-4*x^3), including a(0)=1. - R. J. Mathar, Dec 08 2017
EXAMPLE
a(6) = 123 since M^6 * [1 0 0] = [123 98 76].
a(6) = 123 = 49 + 2*19 + 4*9 = a(5) + 2*a(4) + 4*a(3).
MATHEMATICA
nn=20; a=(1-(2x)^3)/(1-2x); b=x (1-(2x)^3)/(1-2x); CoefficientList[Series[a/(1-b), {x, 0, nn}], x] (* Geoffrey Critzer, Feb 01 2012 *)
LinearRecurrence[{1, 2, 4}, {1, 3, 9}, 40] (* Harvey P. Dale, Nov 02 2016 *)
PROG
(PARI) a(n)=([0, 1, 0; 0, 0, 1; 4, 2, 1]^(n-1)*[1; 3; 9])[1, 1] \\ Charles R Greathouse IV, Feb 28 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Dec 23 2004
STATUS
approved