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%I #61 Feb 17 2024 15:03:32
%S 0,3,15,57,195,633,1995,6177,18915,57513,174075,525297,1582035,
%T 4758393,14299755,42948417,128943555,387027273,1161475035,3485211537,
%U 10457207475,31374768153,94130595915,282404370657,847238277795,2541765165033,7625396158395,22876389801777,68629572058515
%N Total number of different letters summed over all ternary words of length n.
%C These are the numbers d(n,3) studied by J. L. Martin. - _N. J. A. Sloane_, Sep 13 2014
%C 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
%H Philippe Flajolet and Robert Sedgewick, <a href="https://ac.cs.princeton.edu/30mgf/">Combinatorial Parameters and MGFs</a>, lecture slides Analytic Combinatorics, 2012.
%H J. L. Martin, <a href="http://www.math.umn.edu/math/slopes.pdf">The slopes determined by n points in the plane</a> [Dead link]
%H Jeremy L. Martin, <a href="https://arxiv.org/abs/math/0302106">The slopes determined by n points in the plane</a>, 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.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6).
%F E.g.f.: 3*exp(3x) - 3*exp(2x).
%F See Mathematica code for a more transparent version of the e.g.f.
%F Generally for an m-ary word of length n: m*exp(m*x) - m*exp((m-1)*x)
%F From _Alois P. Heinz_, Jan 20 2013: (Start)
%F a(n) = 3*(3^n-2^n) = 3*A001047(n).
%F G.f.: 3*x/((3*x-1)*(2*x-1)).
%F (End)
%F a(n) = A217764(n,5). - _Ross La Haye_, Mar 27 2013
%e 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.
%p a:= n-> 3*(3^n-2^n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jan 20 2013
%t nn=28; Range[0,nn]!CoefficientList[Series[D[(1+y(Exp[x]-1))^3,y]/.y->1, {x,0,nn}], x]
%t (* Second program: *)
%t LinearRecurrence[{5, -6}, {0, 3}, 30] (* _Jean-François Alcover_, Jan 09 2019 *)
%Y Cf. A000918, A001047, A217764.
%Y A diagonal of the triangle in A079268.
%K nonn,easy
%O 0,2
%A _Geoffrey Critzer_, Jan 20 2013
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