A101169 Lengths of successive words (starting with a) under the substitution: {a -> aab, b -> aac, c -> ab}.

1, 3, 9, 26, 76, 221, 644, 1875, 5461, 15903, 46314, 134876, 392791, 1143896, 3331289, 9701475, 28252921, 82278978, 239615244, 697814501, 2032195756, 5918219771, 17235212309, 50192888175, 146173193506, 425689839228

Offset: 0

Author

Jeroen F.J. Laros, Jan 22 2005

Keywords

• easy
• nonn

Examples

			a => aab => aabaabaac => aabaabaacaabaabaacaabaabab, thus a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 26.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2, 3, -1).

Crossrefs

Cf. A101399, A101400.

Programs

Maple

a:= n-> (<<0|1|0>, <0|0|1>, <-1|3|2>>^n. <<0, 1, 3>>)[2,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, May 06 2011

Mathematica

LinearRecurrence[{2,3,-1},{1,3,9},40] (* Harvey P. Dale, Feb 19 2012 *)

Formula

a(n) = 2*a(n-1) + 3*a(n-2) - a(n-3).

G.f.: -(x+1) / (-x^3+3*x^2+2*x-1).

Extensions

Terms >=1875 by Alois P. Heinz, May 06 2011