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A052527
Expansion of (1-x)/(1-x-x^2-x^3+x^4).
2
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
0,4
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
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).
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 *)
PROG
(PARI) my(x='x+O('x^40)); Vec((1-x)/(1-x-x^2-x^3+x^4)) \\ G. C. Greubel, May 13 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (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
(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
CROSSREFS
Sequence in context: A126291 A056224 A293674 * A335443 A042982 A340249
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 05 2000
STATUS
approved