A120908 Sum of the lengths of the drops in all ternary words of length n on {0,1,2}. The drops of a ternary word on {0,1,2} are the subwords 10,20 and 21, their lengths being the differences 1, 2 and 1, respectively.

0, 4, 24, 108, 432, 1620, 5832, 20412, 69984, 236196, 787320, 2598156, 8503056, 27634932, 89282088, 286978140, 918330048, 2927177028, 9298091736, 29443957164, 92980917360, 292889889684, 920511081864, 2887057484028

Offset

1,2

Comments

a(n) = 4*A027471(n).

a(n) = Sum_{k>=0} k*A120907(n,k).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..400

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Index entries for linear recurrences with constant coefficients, signature (6,-9).

Formula

a(n) = 4*(n-1)*3^(n-2).

G.f.: 4*z^2/(1-3*z)^2.

Example

a(2)=4 because the ternary words 00,01,02,11,12 and 22 have no drops, each of the words 10 and 21 has one drop of length 1 and the word 20 has one drop of length 2.

Maple

seq(4*(n-1)*3^(n-2), n=1..27);

Mathematica

Table[4*(n-1)*3^(n-2), {n, 30}] (* Wesley Ivan Hurt, Jan 28 2014 *)
LinearRecurrence[{6, -9}, {0, 4}, 30] (* Harvey P. Dale, Jul 14 2023 *)

Prog

(Magma) [4*(n-1)*3^(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 09 2011
(PARI) a(n) = 4*(n-1)*3^(n-2); \\ Altug Alkan, May 16 2018

Crossrefs

Cf. A027471, A120906, A120907.

Sequence in context: A165752 A366603 A166036 * A145655 A265975 A306610

Adjacent sequences: A120905 A120906 A120907 * A120909 A120910 A120911

Keyword

nonn,easy

Author

Emeric Deutsch, Jul 15 2006

Status

approved