A083944 - OEIS

%I #24 Sep 08 2022 08:45:10

%S 0,1,-2,-3,-10,-19,-42,-83,-170,-339,-682,-1363,-2730,-5459,-10922,

%T -21843,-43690,-87379,-174762,-349523,-699050,-1398099,-2796202,

%U -5592403,-11184810,-22369619,-44739242,-89478483,-178956970,-357913939,-715827882,-1431655763

%N A generalized Jacobsthal sequence.

%H Vincenzo Librandi, <a href="/A083944/b083944.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).

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

%F E.g.f.: (9*exp(x) - 4*exp(2*x) - 5*exp(-x))/6.

%F a(n) = (9 - 2^(n+2) - 5*(-1)^n)/6.

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

%F a(2*n) = -A000975(2*n); a(2*n+1) = 2 - A000975(2*n+1).

%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), with a(0)=0, a(1)=1, a(2)=-2. - _Harvey P. Dale_, Jun 08 2014

%t CoefficientList[Series[x (1-4x)/((1+x)(1-x)(1-2x)),{x,0,40}],x] (* _Vincenzo Librandi_, Apr 04 2012 *)

%t LinearRecurrence[{2,1,-2},{0,1,-2},40] (* _Harvey P. Dale_, Jun 08 2014 *)

%o (Magma) [3/2-2^(n+1)/3-5*(-1)^n/6: n in [0..40]]; // _Vincenzo Librandi_, Apr 04 2012

%o (PARI) x='x+O('x^50); concat([0], Vec(x*(1-4*x)/((1+x)*(1-x)*(1-2*x)))) \\ _G. C. Greubel_, Oct 10 2017

%Y Cf. A083943.

%K sign,easy

%O 0,3

%A _Paul Barry_, May 09 2003