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A084059 a(n) = 4*a(n-1) + 2*a(n-2) for n>1, a(0)=1, a(1)=2. 8

%I

%S 1,2,10,44,196,872,3880,17264,76816,341792,1520800,6766784,30108736,

%T 133968512,596091520,2652303104,11801395456,52510188032,233643543040,

%U 1039594548224,4625665278976,20581850212352,91578731407360

%N a(n) = 4*a(n-1) + 2*a(n-2) for n>1, a(0)=1, a(1)=2.

%C Binomial transform of A002533.

%C 2*A084059 is the Lucas sequence V(4,-2). - _Bruno Berselli_, Jan 09 2013

%H G. C. Greubel, <a href="/A084059/b084059.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 (4,2).

%F E.g.f.: exp(2*x)*cosh(sqrt(6)*x).

%F a(n) = ((2+sqrt(6))^n + (2-sqrt(6))^n)/2. - _Paul Barry_, May 13 2003

%F a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*2^(n-k)*3^k. - _Paul Barry_, Jan 15 2007

%F G.f.: (1-2*x)/(1-4*x-2*x^2). - _Philippe Deléham_, Sep 07 2009

%F a(n) = A090017(n+1) - 2*A090017(n). - _R. J. Mathar_, Apr 05 2011

%F a(n) = Sum_{k=0..n} A201730(n,k)*5^k. - _Philippe Deléham_, Dec 06 2011

%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k-2)/(x*(3*k+1) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 27 2013

%F a(n) = (-i)^n*2^(n/2)*ChebyshevT(n, i*sqrt(2)) = 2^((n-2)/2)*Lucas(n, 2*sqrt(2)). - _G. C. Greubel_, Jan 03 2020

%p seq(simplify(2^(n/2)*(-I)^n*ChebyshevT(n, I*sqrt(2))), n = 0..30); # _G. C. Greubel_, Jan 03 2020

%t Table[(-I)^n*2^(n/2)*ChebyshevT[n, I*Sqrt[2]], {n,0,30}] (* _G. C. Greubel_, Jan 03 2020 *)

%o (Sage) [lucas_number2(n,4,-2)/2 for n in range(0, 30)] # _Zerinvary Lajos_, May 14 2009

%o (MAGMA) [n le 2 select n else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Apr 05 2011

%o (PARI) Vec((1-2*x)/(1-4*x-2*x^2) + O(x^30)) \\ _Michel Marcus_, Feb 04 2016

%o (PARI) vector(31, n, round((-I)^(n-1)*2^((n-1)/2)*polchebyshev(n-1, 1, I*sqrt(2))) ) \\ _G. C. Greubel_, Jan 03 2020

%o (GAP) a:=[1,2];; for n in [3..30] do a[n]:=4*a[n-1]+2*a[n-2]; od; a; # _G. C. Greubel_, Jan 03 2020

%Y Cf. A090017.

%K nonn,easy

%O 0,2

%A _Paul Barry_, May 10 2003

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Last modified January 24 07:18 EST 2020. Contains 331189 sequences. (Running on oeis4.)