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A180343 a(0)=-4; a(n+1) = 2*a(n) + period 4: repeat 6,1,2,5.

%I A180343 #25 Feb 20 2025 12:53:26
%S A180343 -4,-2,-3,-4,-3,0,1,4,13,32,65,132,269,544,1089,2180,4365,8736,17473,
%T A180343 34948,69901,139808,279617,559236,1118477,2236960,4473921,8947844,
%U A180343 17895693,35791392,71582785,143165572,286331149,572662304,1145324609,2290649220,4581298445
%N A180343 a(0)=-4; a(n+1) = 2*a(n) + period 4: repeat 6,1,2,5.
%C A180343 Period 4:repeat 6,1,2,5 = A131800(n-1).
%H A180343 Vincenzo Librandi, <a href="/A180343/b180343.txt">Table of n, a(n) for n = 0..1000</a>
%H A180343 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,1,-2).
%F A180343 G.f.: ( -4 + 6*x + x^2 + 2*x^3 + 9*x^4 ) / ( (x-1)*(2*x-1)*(1+x)*(x^2+1) ). - _R. J. Mathar_, Jan 18 2011
%F A180343 a(n) = 2*a(n-1) + A131800(n+2).
%F A180343 a(n) = a(n-4) + 2^n.
%F A180343 a(n) = a(n-2) + 4*A007909(n) (A007909(0)=0). From second -3.
%F A180343 a(n) = -2*A112030(n+1)/5 - (-1)^n/6 - 7/2 + 2^n/15. - _R. J. Mathar_, Jan 18 2011
%F A180343 a(n) = 2*a(n-1) + a(n-4) - 2*a(n-5). - _Vincenzo Librandi_, Jun 17 2012
%e A180343 a(1) = 2*(-4) + 6 = -2;
%e A180343 a(2) = 2*(-2) + 1 = -3;
%e A180343 a(3) = 2*(-3) + 2 = -4;
%e A180343 a(4) = 2*(-4) + 5 = -3;
%e A180343 a(5) = 2*(-3) + 6 =  0.
%p A180343 A112030 := proc(n) (2+(-1)^n)*(-1)^floor(n/2) ; end proc:
%p A180343 A180343 := proc(n) -2/5*A112030(n+1)-(-1)^n/6-7/2+2^n/15 ; end proc: # _R. J. Mathar_, Jan 18 2011
%t A180343 CoefficientList[Series[(-4+6*x+x^2+2*x^3+9*x^4)/((x-1)*(2*x-1)*(1+x)*(x^2+1)),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 17 2012 *)
%t A180343 LinearRecurrence[{2,0,0,1,-2},{-4,-2,-3,-4,-3},40] (* _Harvey P. Dale_, Sep 06 2020 *)
%o A180343 (Magma)I:=[-4, -2, -3, -4, -3]; [n le 5 select I[n] else 2*Self(n-1)+Self(n-4)-2*Self(n-5): n in [1..40]]; // _Vincenzo Librandi_, Jun 17 2012
%K A180343 sign,easy,less
%O A180343 0,1
%A A180343 _Paul Curtz_, Jan 18 2011