A389989 a(n) = 3^n - 2^n - 2*n - 1, n > 1; a(0)=a(1)=0.

0, 0, 0, 12, 56, 200, 652, 2044, 6288, 19152, 58004, 175076, 527320, 1586104, 4766556, 14316108, 42981152, 129009056, 387158308, 1161737140, 3485735784, 10458256008, 31376865260, 94134790172, 282412759216, 847255054960, 2541798719412, 7625463267204, 22876524019448, 68629840493912

Offset

0,4

Comments

Number of ternary strings of length n such that the number of 0's and 1's in total is at least two and there is at least one 2. For example, a(3)=12 since the strings are (number of permutations in parentheses): 002 (3), 112 (3), 012 (6).

Also the number of ordered set partitions of an n-set into 3 sets such that the number of elements in the first and second sets in total is at least two and the third set has at least one element.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (7,-17,17,-6).

Formula

E.g.f.: (exp(x)-1)*(exp(2*x)-2*x-1).

G.f.: 4*x^3*(3 - 7*x + 3*x^2)/((1 - x)^2*(1 - 2*x)*(1 - 3*x)). - Andrew Howroyd, Nov 06 2025

Example

a(3)=12 since the ordered set partitions are: {1,2}{ }{3}, {1,3}{ }{2}, {2,3}{ }{1}, { }{1,2}{3}, { }{1,3}{2}, { }{2,3}{1}, {1}{2}{3}, {1}{3}{2}, {2}{1}{3}, {2}{3}{1}, {3}{1}{2}, {3}{2}{1}.

Mathematica

a[n_] := 3^n - 2^n - 2*n - 1; a[0] = a[1] = 0; Array[a, 30, 0] (* Amiram Eldar, Oct 21 2025 *)

Crossrefs

Sequence in context: A009430 A348584 A035289 * A275505 A009827 A068418

Adjacent sequences: A389986 A389987 A389988 * A389990 A389991 A389992

Keyword

nonn,easy

Author

Enrique Navarrete, Oct 21 2025

Status

approved