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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.
10
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
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
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
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jul 15 2006
STATUS
approved