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

Author

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

Keywords

• nonn
• easy

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

Links

• P. Braendeen and T. Mansour, Finite automata and pattern avoidance in words
• Tosic R., Masulovic D., Stojmenovic I., Brunvoll J., Cyvin B. N. and Cyvin S. J., 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).

Crossrefs

Cf. A134247, A000124, A144333.

Programs

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 *)

PARI

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

Formula

From Paul Barry, Jul 22 2004: (Start)

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

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

From Paul Barry, Mar 27 2007: (Start)

E.g.f.: e^(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

Extensions

Corrected and extended by Wouter Meeussen, Jul 30 2002

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

New name using explicit formula. - Joerg Arndt, May 21 2018