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A220753 Expansion of (1+4*x+5*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)). 0

%I #19 Sep 08 2022 08:46:04

%S 1,4,8,11,22,25,50,53,106,109,218,221,442,445,890,893,1786,1789,3578,

%T 3581,7162,7165,14330,14333,28666,28669,57338,57341,114682,114685,

%U 229370,229373,458746,458749,917498,917501,1835002,1835005,3670010,3670013

%N Expansion of (1+4*x+5*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-2).

%F G.f.: (1+4*x+5*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)).

%F a(2n) = 7*2^n - 6 = A048489(n) = A063757(2n) = A005009(n)-6.

%F a(2n+1) = 7*2^n - 3 = A048489(n) + 3 = A063757(2n+1) - 3*A000225(n) = A005009(n)-3.

%F a(n) = a(n-1)*2 if n even.

%F a(n) = a(n-1)+3 if n odd.

%F a(n) = 3*a(n-2) - 2*a(n-4) with a(0)=1, a(1)=4, a(2)=8, a(3)=11.

%F a(n) = 7*2^floor(n/2) - (3/2)*(3+(-1)^n).

%F a(n) = A047290(A083416(n+1)). [_Bruno Berselli_, Apr 13 2013]

%t Table[7 2^Floor[n/2] - (3/2) (3 + (-1)^n), {n, 0, 40}] (* _Bruno Berselli_, Apr 13 2013 *)

%t LinearRecurrence[{0, 3, 0, -2}, {1, 4, 8, 11}, 40] (* _T. D. Noe_, Apr 17 2013 *)

%o (Magma) m:=41; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+4*x+5*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)))); // _Bruno Berselli_, Apr 13 2013

%Y Cf. A005009, A048489, A063757.

%K nonn,easy

%O 0,2

%A _Philippe Deléham_, Apr 13 2013

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