A102000 -id:A102000 - cp's OEIS frontend

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

Gary W. Adamson, Dec 23 2004

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

			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).

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).

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 *)

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

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

A102002 Weighted tribonacci (1,2,4), companion to A102001.

Original entry on oeis.org

1, 7, 13, 31, 85, 199, 493, 1231, 3013, 7447, 18397, 45343, 111925, 276199, 681421, 1681519, 4149157, 10237879, 25262269, 62334655, 153810709, 379529095, 936489133, 2310790159, 5701884805, 14069421655, 34716351901, 85662734431, 211373124853, 521564001319

Offset: 1

Gary W. Adamson, Dec 23 2004

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. A102001 is generated from [1 1 1 / 2 0 0 / 0 2 0] but has the same characteristic polynomial and recursive multipliers (1,2,4). A101000 uses the recursive multipliers (1,2,4,8).

			a(6) = 199 = 85 + 2*31 + 4*13 = a(5) + 2*a(4) + 4*a(3).
a(6) = 199 since M^6 * [1 1 1] = [85 199 493] = [a(5) a(6) a(7)].

Index entries for linear recurrences with constant coefficients, signature (1,2,4).

Cf. A102000, A102001.

LinearRecurrence[{1,2,4}, {1,7,13}, 50] (* Harvey P. Dale, Apr 28 2012 *)

from sage.combinat.sloane_functions import recur_gen3
it = recur_gen3(1,1,1,1,2,4)
[next(it) for i in range(32)]
# Zerinvary Lajos, Jun 25 2008

a(n) = a(n-1) + 2*a(n-2) + 4*a(n-3), a>3. a(n) = center term in M^n * [1 1 1], where M = the 3X3 matrix [0 1 0 / 0 0 1 / 4 2 1]; M^n * [1 1 1] = [a(n-1) a(n) a(n+1)].

G.f.: -x*(4*x^2+6*x+1)/(4*x^3+2*x^2+x-1). [Harvey P. Dale, Apr 28 2012]

More terms from Harvey P. Dale, Apr 28 2012

cp's OEIS Frontend

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

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