A210448 Total number of different letters summed over all ternary words of length n.

0, 3, 15, 57, 195, 633, 1995, 6177, 18915, 57513, 174075, 525297, 1582035, 4758393, 14299755, 42948417, 128943555, 387027273, 1161475035, 3485211537, 10457207475, 31374768153, 94130595915, 282404370657, 847238277795, 2541765165033, 7625396158395, 22876389801777, 68629572058515

Offset

0,2

Comments

These are the numbers d(n,3) studied by J. L. Martin. - N. J. A. Sloane, Sep 13 2014

For n >= 0, the number of ternary sequences of length n+1, that contain at least one pair of same consecutive digits. - Armend Shabani, Apr 10 2019

Links

Table of n, a(n) for n=0..28.

Philippe Flajolet and Robert Sedgewick, Combinatorial Parameters and MGFs, lecture slides Analytic Combinatorics, 2012.

J. L. Martin, The slopes determined by n points in the plane [Dead link]

Jeremy L. Martin, The slopes determined by n points in the plane, arXiv:math/0302106 [math.AG], 2003-2006; Duke Math. J. 131 (2006), no. 1, 119-165. See table of d(n,k), but beware errors.

Index entries for linear recurrences with constant coefficients, signature (5,-6).

Formula

E.g.f.: 3*exp(3x) - 3*exp(2x).

See Mathematica code for a more transparent version of the e.g.f.

Generally for an m-ary word of length n: m*exp(m*x) - m*exp((m-1)*x)

From Alois P. Heinz, Jan 20 2013: (Start)

a(n) = 3*(3^n-2^n) = 3*A001047(n).

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

(End)

a(n) = A217764(n,5). - Ross La Haye, Mar 27 2013

Example

a(2) = 15 because the length 2 words on alphabet {0,1,2} are: 00, 01, 02, 10, 11, 12, 20, 21, 22 and we sum respectively 1 + 2 + 2 + 2 + 1 + 2 + 2 + 2 + 1 = 15.

Maple

a:= n-> 3*(3^n-2^n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 20 2013

Mathematica

nn=28; Range[0, nn]!CoefficientList[Series[D[(1+y(Exp[x]-1))^3, y]/.y->1, {x, 0, nn}], x]
(* Second program: *)
LinearRecurrence[{5, -6}, {0, 3}, 30] (* Jean-François Alcover, Jan 09 2019 *)

Crossrefs

Cf. A000918, A001047, A217764.

A diagonal of the triangle in A079268.

Sequence in context: A049161 A358685 A118048 * A284699 A218657 A218804

Adjacent sequences: A210445 A210446 A210447 * A210449 A210450 A210451

Keyword

nonn,easy

Author

Geoffrey Critzer, Jan 20 2013

Status

approved