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

Author

Geoffrey Critzer, Jan 20 2013

Keywords

• nonn
• easy

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

Examples

			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.
		

Links

• 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).

Crossrefs

Cf. A000918, A001047, A217764.

A diagonal of the triangle in A079268.

Programs

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 *)

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