A102001 A weighted tribonacci, (1,2,4).

1, 3, 9, 19, 49, 123, 297, 739, 1825, 4491, 11097, 27379, 67537, 166683, 411273, 1014787, 2504065, 6178731, 15246009, 37619731, 92826673, 229050171, 565182441, 1394589475, 3441155041, 8491063755, 20951731737, 51698479411, 127566197905, 314770083675

Offset: 1

Author

Gary W. Adamson, Dec 23 2004

Keywords

• nonn
• easy

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

Examples

			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).
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 19.
• Index entries for linear recurrences with constant coefficients, signature (1,2,4).

Programs

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 *)

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

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