A078010 Expansion of (1-x)/(1 - x - x^2 - 2*x^3).

1, 0, 1, 3, 4, 9, 19, 36, 73, 147, 292, 585, 1171, 2340, 4681, 9363, 18724, 37449, 74899, 149796, 299593, 599187, 1198372, 2396745, 4793491, 9586980, 19173961, 38347923, 76695844, 153391689, 306783379, 613566756, 1227133513, 2454267027, 4908534052

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• nonn
• easy

Comments

For n > 0, a(n) is the number of ways to tile a strip of length n with squares, dominoes, and two colors of trominoes, with the restriction that the first tile cannot be a square. - Greg Dresden and Bora Bursalı, Aug 31 2023

Examples

			a(6) = 19 = A077947(4) + 2*A077947(3) = 9 + 2*5 = 19.
G.f. = 1 + x^2 + 3*x^3 + 4*x^4 + 9*x^5 + 19*x^6 + 36*x^7 + 73*x^8 + ... - _Michael Somos_, Nov 18 2020
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Crossrefs

Cf. A033138, A077947.

Programs

GAP

a:=[1,0,1];; for n in [4..50] do a[n]:=a[n-1]+a[n-2]+2*a[n-3]; od; a; # G. C. Greubel, Jun 28 2019

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x-x^2-2*x^3) )); // G. C. Greubel, Jun 28 2019

Mathematica

CoefficientList[Series[(1-x)/(1-x-x^2-2*x^3), {x,0,50}],x]  (* Harvey P. Dale, Mar 17 2011 *)
LinearRecurrence[{1, 1, 2}, {1, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)

PARI

Vec((1-x)/(1-x-x^2-2*x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 26 2012

PARI

{a(n) = ([0, 1, 1; 1, 1, 0; 0, 2, 0]^n)[1, 1]}; /* Michael Somos, Nov 18 2020 */

Sage

((1-x)/(1-x-x^2-2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 28 2019

Formula

a(0)=1, a(1)=0, a(2)=1, a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n > 2. - Philippe Deléham, Sep 19 2006

a(n) + a(n+1) = A122552(n+1). - Philippe Deléham, Sep 25 2006

If p[1]=0, p[2]=1, p[i]=3, (i>2), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n >= 1, a(n)=det A. - Milan Janjic, May 02 2010

For n > 3, a(n) = A077947(n-2) + 2*A077947(n-3), with A077947 beginning (1, 2, 5, 9, 18, 37, ...); "1" has offset 1. - Gary W. Adamson, May 13 2013

a(n) = 2^(n-1) - 3*floor((2^(n-1))/7) - 1, for n >= 1. - Ridouane Oudra, Dec 02 2019

G.f.: (1 - x) / ((1 - 2*x) * (1 + x + x^2)). - Michael Somos, Nov 18 2020