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A052545
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Expansion of (1-x)^2/(1-3*x+x^3).
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4
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1, 1, 4, 11, 32, 92, 265, 763, 2197, 6326, 18215, 52448, 151018, 434839, 1252069, 3605189, 10380728, 29890115, 86065156, 247814740, 713554105, 2054597159, 5915976737, 17034376106, 49048531159, 141229616740, 406654474114
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OFFSET
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0,3
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COMMENTS
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(1, 4, 11, 32, ...) = INVERT transform of (1, 3, 4, 5, 6, 7, ...).
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LINKS
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FORMULA
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G.f.: (1-x)^2/(1-3*x+x^3).
a(n) = 3*a(n-1) - a(n-3), with a(0)=a(1)=1, a(2)=4.
a(n) = Sum_{alpha = RootOf(1-3*x+x^3)} (-1/9 * (-1+2*alpha^2-2*alpha) * alpha^(-1-n)).
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MAPLE
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spec := [S, {S=Sequence(Prod(Z, Union(Z, Sequence(Z)), Sequence(Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
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LinearRecurrence[{3, 0, -1}, {1, 1, 4}, 40] (* G. C. Greubel, May 08 2019 *)
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PROG
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(Python)
TOP = 33
a = [1]*TOP
a[2]=4
for n in range(3, TOP):
print(a[n-3], end=', ')
a[n] = 3*a[n-1] - a[n-3]
(PARI) my(x='x+O('x^40)); Vec((1-x)^2/(1-3*x+x^3)) \\ G. C. Greubel, May 08 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)^2/(1-3*x+x^3) )); // G. C. Greubel, May 08 2019
(Sage) ((1-x)^2/(1-3*x+x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 08 2019
(GAP) a:=[1, 1, 4];; for n in [4..40] do a[n]:=3*a[n-1]-a[n-3]; od; a; # G. C. Greubel, May 08 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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STATUS
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approved
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