A072863 a(n) = 2^(n-3)*(n^2+3*n+8).

1, 3, 9, 26, 72, 192, 496, 1248, 3072, 7424, 17664, 41472, 96256, 221184, 503808, 1138688, 2555904, 5701632, 12648448, 27918336, 61341696, 134217728, 292552704, 635437056, 1375731712, 2969567232, 6392119296, 13723762688

Offset

0,2

Comments

Binomial transform of 1+n*(n+1)/2, A000124.

Number of 123-avoiding ternary words of length n-1.

Row sums of triangle A134247. Also double binomial transform of (1, 1, 1, 0, 0, 0, ...). - Gary W. Adamson, Oct 15 2007

Equals row sums of triangle A144333. - Gary W. Adamson, Sep 18 2008

From Enrique Navarrete, Nov 22 2025: (Start)

Partial sums of A190050.

Number of ternary strings of length n with at most two 0's. (End)

Links

Table of n, a(n) for n=0..27.

Petter Brändén and Toufik Mansour, Finite automata and pattern avoidance in words, arXiv:math/0309269 [math.CO], 2003.

Ratko Tosic, Dragan Masulovic, Ivan Stojmenovic, Jon Brunvoll, Bjorg N. Cyvin, and Sven J. Cyvin, Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., 1995, 35, 181-187, Table 1, with an error at h=16.

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Formula

From Paul Barry, Jul 22 2004: (Start)

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

a(n) = 2^(n-3)*(n^2+3*n+8). (End)

From Paul Barry, Mar 27 2007: (Start)

E.g.f.: exp(2*x)*(1+x+x^2/2);

a(n) = Sum_{k=0..2} binomial(n,k)*2^(n-k). (End)

a(n-1) + A001788(n-2) = A104270(n). - R. J. Mathar, May 21 2018

From Enrique Navarrete, Nov 22 2025: (Start)

a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3).

a(n) = 3^n - A345954(n). (End)

Maple

A072863 := proc(n)
    2^(n-3)*(n^2+3*n+8) ;
end proc: # R. J. Mathar, May 21 2018

Mathematica

Table[Sum[Binomial[m-1, k](#^2/2 -#/2 +1 &)[k+1], {k, 0, m}], {m, 36}]
LinearRecurrence[{6, -12, 8}, {1, 3, 9}, 30] (* Harvey P. Dale, May 15 2019 *)

Prog

(PARI) a(n)=2^(n-3)*(n^2+3*n+8); \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Cf. A134247, A000124, A144333.

Cf. A190050, A345954.

Sequence in context: A048470 A138237 A121286 * A054963 A291733 A118046

Adjacent sequences: A072860 A072861 A072862 * A072864 A072865 A072866

Keyword

nonn,easy

Author

Michael A. Childers (childers_moof(AT)yahoo.com), Jul 27 2002

Extensions

Corrected and extended by Wouter Meeussen, Jul 30 2002

Title and offset corrected by R. J. Mathar, May 21 2018

New name using explicit formula by Joerg Arndt, May 21 2018

Status

approved