A135530 - OEIS

%I #30 Feb 03 2024 10:17:00

%S 2,1,4,2,8,4,16,8,32,16,64,32,128,64,256,128,512,256,1024,512,2048,

%T 1024,4096,2048,8192,4096,16384,8192,32768,16384,65536,32768,131072,

%U 65536,262144,131072,524288,262144,1048576

%N a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3), with a(0)=2, a(1)=1.

%H G. C. Greubel, <a href="/A135530/b135530.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 (0,2).

%F From _R. J. Mathar_, Feb 23 2008: (Start)

%F O.g.f.: -(2+x)/(2*x^2-1).

%F a(n) = 2*a(n-2).

%F a(n) = A077957(n+1) + A077957(n+2). (End)

%F E.g.f.: (1/sqrt(2))*( 2*sqrt(2)*cosh(sqrt(2)*x) + sinh(sqrt(2)*x) ). - _G. C. Greubel_, Oct 17 2016

%F a(n) = A076736(n+4) for n >= 0. - _Georg Fischer_, Nov 03 2018

%F From _Amiram Eldar_, Feb 02 2024: (Start)

%F Sum_{n>=0} 1/a(n) = 3.

%F Sum_{n>=0} (-1)^(n+1)/a(n) = 1. (End)

%t CoefficientList[Series[(-x-2)/(2x^2-1),{x,0,40}],x]

%t Transpose[NestList[{#[[2]],Last[#],Last[#]+2#[[2]]-2First[#]}&,{2,1,4},45]][[1]]  (* _Harvey P. Dale_, Mar 05 2011 *)

%t LinearRecurrence[{0, 2}, {2, 1}, 25] (* _G. C. Greubel_, Oct 17 2016 *)

%o (PARI) a(n)=1<<(1-n%2+n\2) \\ _Charles R Greathouse IV_, Jun 01 2011

%Y Cf. A076736, A077957.

%K nonn,easy

%O 0,1

%A _Paul Curtz_, Feb 20 2008

%E More terms from _R. J. Mathar_, Feb 23 2008