%I #29 Dec 08 2017 11:26:10

%S 1,3,9,19,49,123,297,739,1825,4491,11097,27379,67537,166683,411273,

%T 1014787,2504065,6178731,15246009,37619731,92826673,229050171,

%U 565182441,1394589475,3441155041,8491063755,20951731737,51698479411,127566197905,314770083675

%N A weighted tribonacci, (1,2,4).

%C 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.

%C 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

%C 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

%H Harvey P. Dale, <a href="/A102001/b102001.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,4).

%F 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].

%F a(n) = Sum{k=0..n} T(n-k, k)2^k, T(n, k) = trinomial coefficients (A027907). - _Paul Barry_, Feb 15 2005

%F 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

%F G.f.: 1/(1-x-2*x^2-4*x^3), including a(0)=1. - _R. J. Mathar_, Dec 08 2017

%e a(6) = 123 since M^6 * [1 0 0] = [123 98 76].

%e a(6) = 123 = 49 + 2*19 + 4*9 = a(5) + 2*a(4) + 4*a(3).

%t 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 *)

%t LinearRecurrence[{1,2,4},{1,3,9},40] (* _Harvey P. Dale_, Nov 02 2016 *)

%o (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

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Dec 23 2004