A004054 Expansion of (1-x)/((1+x)*(1-2*x)*(1-3*x)).

1, 3, 11, 35, 111, 343, 1051, 3195, 9671, 29183, 87891, 264355, 794431, 2386023, 7163531, 21501515, 64526391, 193622863, 580955971, 1743042675, 5229477551, 15689131703, 47068793211, 141209175835, 423633119911, 1270910544543, 3812754003251, 11438306748995

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of paths with n+2 steps on the cycle graph C_6 which start at the first node and end at the 3rd node and each step is -1, 0 or +1. - Herbert Kociemba, Sep 30 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Xavier Acloque, Polynexus Numbers and other mathematical wonders.
• Index entries for linear recurrences with constant coefficients, signature (4,-1,-6).

Crossrefs

Cf. A001045, A001047.

Cf. A000244, A078008.

Programs

Magma

[Ceiling(3^(n+2)/6+(-1)^(n+2)/6-0^n/6-2^(n+2)/6) : n in [0..30]]; // Vincenzo Librandi, Oct 08 2011

Mathematica

Table[1/6 ((-1)^(2+n)-2^(n+2)+3^(n+2)),{n,0,30}] (* Herbert Kociemba, Sep 30 2020 *)

PARI

Vec((1-x)/((1+x)*(1-2*x)*(1-3*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

From Paul Barry, Sep 13 2003: (Start)

The sequence 0, 0, 1, ... has a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*A001045(2*k).

a(n) = 3^n/6 + (-1)^n/6 - 0^n/6 - 2^n/6. (End)

The signed sequence 0, 1, -3, ... has g.f. x*(1+x)/((1-x)*(1+2*x)*(1+3*x)) and a(n) = 1/6 + (-2)^n/3 - (-3)^n/2. It is the third inverse binomial transform of A001045(2*n-1) - 0^n/2. - Paul Barry, Apr 21 2004

From Paul Barry, Jul 22 2004: (Start)

Convolution of A000244 and A078008.

a(n) = Sum_{k=0..n} A078008(k)*3^(n-k).

a(n) = (3*A000244(n) - A001045(n+2))/2. (End)

a(n) = (A001047(n+2) + (-1)^n)/6. - Vladimir Pletser, Dec 02 2023

a(n) = A094705(n+1)-A094705(n). - R. J. Mathar, Dec 02 2023