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a(n) = ((3+sqrt(5))*(1+sqrt(5))^n + (3-sqrt(5))*(1-sqrt(5))^n)/2.
3

%I #17 Sep 08 2022 08:45:46

%S 3,8,28,88,288,928,3008,9728,31488,101888,329728,1067008,3452928,

%T 11173888,36159488,117014528,378667008,1225392128,3965452288,

%U 12832473088,41526755328,134383403008,434873827328,1407281266688

%N a(n) = ((3+sqrt(5))*(1+sqrt(5))^n + (3-sqrt(5))*(1-sqrt(5))^n)/2.

%C Binomial transform of A163114. Inverse binomial transform of A163063.

%H G. C. Greubel, <a href="/A163062/b163062.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,4).

%F a(n) = 2*a(n-1) + 4*a(n-2) for n > 1; a(0) = 3, a(1) = 8.

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

%F a(n) = 2^n * A000032(n+2). - _Diego Rattaggi_, Jun 17 2020

%t CoefficientList[Series[(3+2*x)/(1-2*x-4*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{2,4}, {3,8}, 30] (* _G. C. Greubel_, Dec 22 2017 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((3+r)*(1+r)^n+(3-r)*(1-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 21 2009

%o (Magma) I:=[3,8]; [n le 2 select I[n] else 2*Self(n-1) + 4*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 22 2017

%o (PARI) x='x+O('x^30); Vec((3+2*x)/(1-2*x-4*x^2)) \\ _G. C. Greubel_, Dec 22 2017

%Y Cf. A000032, A163063, A163114.

%K nonn

%O 0,1

%A Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Jul 21 2009