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

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

Author

Gary W. Adamson, Dec 23 2004

Keywords

• nonn
• easy

Comments

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).

Examples

			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)].
		

Links

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

Crossrefs

Cf. A102000, A102001.

Programs

Mathematica

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

Sage

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

Formula

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]

Extensions

More terms from Harvey P. Dale, Apr 28 2012