A077995 Expansion of (1 - x)/(1 - 2*x - 2*x^2 - x^3).

1, 1, 4, 11, 31, 88, 249, 705, 1996, 5651, 15999, 45296, 128241, 363073, 1027924, 2910235, 8239391, 23327176, 66043369, 186980481, 529374876, 1498754083, 4243238399, 12013359840, 34011950561, 96293859201, 272624979364, 771849627691, 2185243073311, 6186810381368

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• nonn
• easy

Comments

Equals INVERT transform of (1, 3, 4, 4, 4, ...). - Gary W. Adamson, Jan 03 2009

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,1).

Programs

GAP

a:=[1,1,4];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Jun 27 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/( 1-2*x-2*x^2-x^3) )); // G. C. Greubel, Jun 27 2019

Mathematica

CoefficientList[Series[(1-x)/(1-2x-2x^2-x^3),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,1},{1,1,4},30] (* Harvey P. Dale, Sep 11 2013 *)

Maxima

a(n):=sum(sum(binomial(m+i-1,m-1)*sum(binomial(j,n-3*m+2*j-i) *binomial(m,j) *2^(n-3*m+2*j-i),j,0,m) ,i,0,n-m) ,m,1,n); /* Vladimir Kruchinin, May 12 2011 */

PARI

Vec((1-x)/(1-2*x-2*x^2-x^3)+O(x^30)) \\ Charles R Greathouse IV, Sep 24 2012

Sage

((1-x)/(1-2*x-2*x^2-x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019

Formula

a(n) = Sum_{m=1..n} Sum_{i=0..n-m} C(m+i-1,m-1)*Sum_{j=0..m} C(j,n-3*m +2*j-i) * C(m,j)*2^(n-3*m+2*j-i), n>0, a(0)=1. - Vladimir Kruchinin, May 12 2011

G.f.: 1 + x/(G(0)-x) where G(k) = 1 - x*(2*k+2)/(1 - 1/(1 + (2*k+2)/G(k+1)));(continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 17 2012

a(n) = 2*a(n-1) + 2*a(n-2) + a(n-3); a(0)=1, a(1)=1, a(2)=4. - Harvey P. Dale, Sep 11 2013