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%I #35 Sep 08 2022 08:44:59
%S 1,0,1,2,2,5,8,13,24,40,69,120,205,354,610,1049,1808,3113,5360,9232,
%T 15897,27376,47145,81186,139810,240765,414616,714005,1229576,2117432,
%U 3646397,6279400,10813653,18622018,32068674,55224945,95101984
%N Expansion of (1-x)/(1-x-x^2-x^3+x^4).
%C First differences of A116732 (shifted left 3 places). - _R. J. Mathar_, Nov 27 2011
%C 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
%C 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
%H G. C. Greubel, <a href="/A052527/b052527.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=453">Encyclopedia of Combinatorial Structures 453</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1).
%F G.f.: (1-x)/(1 - x - x^2 - x^3 + x^4).
%F 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.
%F 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).
%p spec := [S,{S=Sequence(Prod(Z,Z,Union(Z,Sequence(Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t LinearRecurrence[{1,1,1,-1}, {1,0,1,2}, 40] (* _G. C. Greubel_, May 13 2019 *)
%o (PARI) my(x='x+O('x^40)); Vec((1-x)/(1-x-x^2-x^3+x^4)) \\ _G. C. Greubel_, May 13 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/(1-x-x^2-x^3+x^4) )); // _G. C. Greubel_, May 13 2019
%o (Sage) ((1-x)/(1-x-x^2-x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, May 13 2019
%o (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
%K easy,nonn
%O 0,4
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 05 2000
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