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a(n) = 2^n - A081374(n).
4

%I #50 Jan 30 2024 17:39:41

%S 1,2,6,11,22,42,85,170,342,683,1366,2730,5461,10922,21846,43691,87382,

%T 174762,349525,699050,1398102,2796203,5592406,11184810,22369621,

%U 44739242,89478486,178956971,357913942,715827882,1431655765,2863311530,5726623062

%N a(n) = 2^n - A081374(n).

%H Vincenzo Librandi, <a href="/A083322/b083322.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,2).

%F G.f.: x*(1+2*x^2) / ( (1-2*x)*(1+x)*(1-x+x^2) ). - _R. J. Mathar_, May 27 2011

%F From _Paul Curtz_, May 27 2011: (Start)

%F a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4).

%F a(n)+a(n+3) = 3*2^(n+1) = A007283(n+1).

%F a(n+6)-a(n) = 21*2^(n+1) = A175805(n+1).

%F (End)

%t CoefficientList[Series[(1 + 2 x^2) / ((1 - 2 x) (1 + x) (1 - x + x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 08 2016 *)

%t LinearRecurrence[{2,0,-1,2},{1,2,6,11},40] (* _Harvey P. Dale_, Jan 30 2024 *)

%o (Magma) I:=[1,2,6,11]; [n le 4 select I[n] else 2*Self(n-1)-Self(n-3)+2*Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jul 08 2016

%Y Cf. A081374.

%Y Trisections: A082365, A007613, A132804.

%K nonn,easy

%O 1,2

%A _David Applegate_, Aug 22 2003