A094388 Expansion of (1- 2*x - x^2)/((1-x)*(1-3*x)).

1, 2, 4, 10, 28, 82, 244, 730, 2188, 6562, 19684, 59050, 177148, 531442, 1594324, 4782970, 14348908, 43046722, 129140164, 387420490, 1162261468, 3486784402, 10460353204, 31381059610, 94143178828, 282429536482, 847288609444

Offset: 0

Author

Paul Barry, Apr 28 2004

Keywords

• easy
• nonn

Comments

Binomial transform of 0^n + A001045(n).

From J. M. Bergot, Nov 10 2012: (Start)

Form an array with the first row and column containing all 1's: m(n,1) = m(1,n) = 1 for n=1,2,3,... An interior term m(i,j) is the sum of all preceding terms in row(i) and all preceding terms in column(j): m(i,j) = Sum_{k=1..j-1} m(i,k) + Sum_{l=1..i-1} m(l,j). The sum of the terms in each antidiagonal will reproduce the terms in this sequence beginning at a(0).

The upper left corner of the array begins

1 1 1 1 1 ...

1 2 4 8 16 ...

1 4 10 24 56 ...

1 8 24 66 172 ...

1 16 56 172 490 ...

...

(End) [edited by Jon E. Schoenfield, Sep 08 2018]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..600
• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Crossrefs

Cf. A001045, A007051, A034472.

Programs

Magma

[3^n/3-0^n/3+1: n in [0..30]]; // Vincenzo Librandi, May 21 2011

Mathematica

CoefficientList[Series[(1-2x-x^2)/((1-x)(1-3x)),{x,0,30}],x] (* Harvey P. Dale, May 20 2011 *)

PARI

a(n)=3^n/3-0^n/3+1 \\ Charles R Greathouse IV, Nov 27 2012

SageMath

[(3^n +3 -int(n==0))//3 for n in range(41)] # G. C. Greubel, Sep 27 2024

Formula

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

a(n+1) = 2*A007051(n).

a(n) = A034472(n-1), n > 0. - R. J. Mathar, Sep 05 2008

G.f.: G(0), where G(k)= 1 + 3^k*x/(1 - x/(x + 3^k*x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2013

E.g.f.: (1/3)*(-1 + 3*exp(x) + exp(3*x)). - G. C. Greubel, Sep 27 2024