A077834 Expansion of 1/(1 - 2*x - 2*x^2 - 3*x^3).

1, 2, 6, 19, 56, 168, 505, 1514, 4542, 13627, 40880, 122640, 367921, 1103762, 3311286, 9933859, 29801576, 89404728, 268214185, 804642554, 2413927662, 7241782987, 21725348960, 65176046880, 195528140641, 586584421922, 1759753265766, 5279259797299, 15837779391896

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (2,2,3)

Programs

Maple

A049347 := proc(n) op(1+(n mod 3),[1,-1,0]) ; end proc:
A077834 := proc(n) (3^(n+2)+3*A049347(n-1)+4*A049347(n))/13 ; end proc:
seq(A077834(n),n=0..20) ; # R. J. Mathar, Mar 22 2011

Mathematica

k0=k1=0;lst={};Do[kt=k1;k1=3^n-k1-k0;k0=kt;AppendTo[lst, k1/3], {n, 1, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
CoefficientList[Series[1/(1-2x-2x^2-3x^3),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,3},{1,2,6},30] (* Harvey P. Dale, Jan 31 2012 *)

PARI

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

Formula

From Paul Barry, May 19 2004: (Start)

Convolution of A000244 and A049347.

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

a(n) = sum_{k=0..n} (3^k*2*sqrt(3)*cos(2*Pi*(n-k)/3 + Pi/6)/3).

a(n) = 3^(n+2)/13 + 2*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/39 + 2*sqrt(3)*sin(2*Pi*n/3 + Pi/3)/13.

(End)

a(n) = A152733(n+3)/3. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

a(0)=1, a(1)=2, a(2)=6, a(n) = 2*a(n-1) + 2*a(n-2) + 3*a(n-3). - Harvey P. Dale, Jan 31 2012

a(n) = 1/52*(4*3^(n + 2) + (-1)^n*(2*(-1)^floor(n/3) + 9*(-1)^floor((1 + n)/3) + 6*(-1)^floor((n + 2)/3) + (-1)^floor((n + 4)/3))). - John M. Campbell, Dec 23 2016