A052527 Expansion of (1-x)/(1-x-x^2-x^3+x^4).

1, 0, 1, 2, 2, 5, 8, 13, 24, 40, 69, 120, 205, 354, 610, 1049, 1808, 3113, 5360, 9232, 15897, 27376, 47145, 81186, 139810, 240765, 414616, 714005, 1229576, 2117432, 3646397, 6279400, 10813653, 18622018, 32068674, 55224945, 95101984

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

First differences of A116732 (shifted left 3 places). - R. J. Mathar, Nov 27 2011

a(n) is the number of ways to tile an n-board (a board with dimensions n X 1) using one type of domino, two types of straight tromino, and one type each of all other straight m-ominoes for m > 3. - Michael A. Allen, Sep 17 2020

Equivalently, a(n) is the number of compositions of n into parts >= 2 where there are two kinds of part 3. - Joerg Arndt, Sep 18 2020

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 453
• Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1).

Programs

GAP

a:=[1,0,1,2];; for n in [5..40] do a[n]:=a[n-1]+a[n-2]+a[n-3] -a[n-4]; od; a; # G. C. Greubel, May 13 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/(1-x-x^2-x^3+x^4) )); // G. C. Greubel, May 13 2019

Maple

spec := [S,{S=Sequence(Prod(Z,Z,Union(Z,Sequence(Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

LinearRecurrence[{1,1,1,-1}, {1,0,1,2}, 40] (* G. C. Greubel, May 13 2019 *)

PARI

my(x='x+O('x^40)); Vec((1-x)/(1-x-x^2-x^3+x^4)) \\ G. C. Greubel, May 13 2019

Sage

((1-x)/(1-x-x^2-x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 13 2019

Formula

G.f.: (1-x)/(1 - x - x^2 - x^3 + x^4).

a(n) = a(n+1) +a(n+2) +a(n+3) -a(n+4), a(0)=1, a(1)=0, a(2)=1, a(3)=2.

a(n) = Sum_{alpha = RootOf(1-x-x^2-x^3+x^4)} (1/39)*(2 + 11*alpha - 4*alpha^2 - alpha^3)*alpha^(-1-n).

Extensions

More terms from James Sellers, Jun 05 2000