A078043 - OEIS

%I #37 Mar 29 2018 09:55:28

%S 1,-2,3,-7,14,-27,55,-110,219,-439,878,-1755,3511,-7022,14043,-28087,

%T 56174,-112347,224695,-449390,898779,-1797559,3595118,-7190235,

%U 14380471,-28760942,57521883,-115043767,230087534,-460175067,920350135,-1840700270,3681400539,-7362801079

%N Expansion of (1 - x)/(1 + x - x^2 + 2*x^3).

%C abs(a(n)) = A033129(n+1) - A033129(n). - _Reinhard Zumkeller_, Feb 22 2010

%H Robert Israel, <a href="/A078043/b078043.txt">Table of n, a(n) for n = 0..3318</a>

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

%F a(0)=1, a(1)=-2, a(2)=3, a(n) = -a(n-1) + a(n-2) - 2*a(n-3). - _Harvey P. Dale_, Feb 02 2012 [corrected by _Wojciech Florek_, Feb 26 2018]

%F a(n) = (1/21) * (-9*2^n*e^(i*n*Pi) + 9*cos((n*Pi)/3) - sqrt(3)*sin((n*Pi)/3)). - _Harvey P. Dale_, Feb 02 2012

%F a(n) = (1/7) * (6*(-2)^n + [1,-2,-3,-1,2,3](mod 6)) = A077972(n) - A077972(n-1). - _Ralf Stephan_, Aug 18 2013

%F a(n) = floor((6*(-2)^n+3)/7). - _Tani Akinari_, Oct 05 2014

%p f:= gfun:-rectoproc({a(0)=1, a(1)=-2, a(2)=3, a(n) = -a(n-1) + a(n-2) - 2*a(n-3)},a(n),remember):

%p map(f, [$0..40]); # _Robert Israel_, Mar 28 2018

%t CoefficientList[Series[(1-x)/(1+x-x^2+2x^3),{x,0,40}],x] (* or *) LinearRecurrence[{-1,1,-2},{1,-2,3},40] (* _Harvey P. Dale_, Feb 02 2012 *)

%o (PARI) Vec((1-x)/(1+x-x^2+2*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (PARI) a(n)=1/7*(6*(-2)^n+[1,-2,-3,-1,2,3][(n%6)+1]) /* _Ralf Stephan_, Aug 18 2013 */

%o (PARI) a(n)=(6*(-2)^n+3)\7 \\ _Tani Akinari_, Oct 05 2014

%K sign,easy

%O 0,2

%A _N. J. A. Sloane_, Nov 17 2002