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A052941
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Expansion of (1-x)/(1 - 4*x + x^2 + x^3).
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3
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1, 3, 11, 40, 146, 533, 1946, 7105, 25941, 94713, 345806, 1262570, 4609761, 16830668, 61450341, 224360935, 819162731, 2990839648, 10919834926, 39869337325, 145566674726, 531477526653, 1940474094561, 7084852176865
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OFFSET
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0,2
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COMMENTS
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a(n) = term (3,1) in M^n, M = the 3 X 3 matrix [1,1,2; 1,2,1; 1,1,1]. - Gary W. Adamson, Mar 12 2009
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LINKS
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FORMULA
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G.f.: (1-x)/(1 - 4*x + x^2 + x^3).
a(n) = 4*a(n-1) - a(n-2) - a(n-3).
a(n) = Sum_{alpha=RootOf(1-4*z+z^2+z^3)} (3-alpha^2)*alpha^(-1-n)/13.
a(n) = (b(n+2) - b(n+1) + b(n))/13, where b(n) = A356463(n). - Ding Hao, Aug 08 2022
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MAPLE
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spec:= [S, {S=Sequence(Union(Z, Z, Prod(Union(Sequence(Z), Z), Z)))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
seq(coeff(series((1-x)/(1-4*x+x^2+x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 18 2019
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MATHEMATICA
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PROG
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(Magma) I:=[1, 3, 11]; [n le 3 select I[n] else 4*Self(n-1)-Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012
(PARI) my(x='x+O('x^30)); Vec((1-x)/(1-4*x+x^2+x^3)) \\ Altug Alkan, Sep 21 2018
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1-x)/(1-4*x+x^2+x^3)).list()
(GAP) a:=[1, 3, 11];; for n in [4..30] do a[n]:=4*a[n-1]-a[n-2]-a[n-3]; od; a; # G. C. Greubel, Oct 18 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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EXTENSIONS
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STATUS
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approved
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