A077860 - OEIS

%I #56 Jan 03 2025 13:31:29

%S 1,3,5,5,1,-7,-15,-15,1,33,65,65,1,-127,-255,-255,1,513,1025,1025,1,

%T -2047,-4095,-4095,1,8193,16385,16385,1,-32767,-65535,-65535,1,131073,

%U 262145,262145,1,-524287,-1048575,-1048575,1,2097153,4194305,4194305,1,-8388607,-16777215,-16777215

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

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,2).

%F a(n) = 1-A146559(n+2). a(n)= 3*a(n-1) -4*a(n-2) +2*a(n-3). - _R. J. Mathar_, Jan 18 2011

%F G.f.: Q(0) where Q(k) = 1 + k*(2*x+1) + 8*x - 2*x*(k+1)*(k+5)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Mar 14 2013

%F G.f.: G(0)/(2*(1-x)^2), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+2) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 25 2013

%F a(n) = Sum_{k=0..n} ((-1)^k*2^(n-k)*binomial(n-k-1,k)). - _Vladimir Kruchinin_, Jul 02 2015

%F a(n) = 1 + 2^(1 + n/2)*sin((n*Pi)/4). - _Jean-François Alcover_, Jul 02 2015

%F a(n) = 1 + 2*Im((1 + i)^n), where i is the imaginary unit. - _Daniel Suteu_, Dec 21 2018

%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n+2,2*k+2). - _Taras Goy_, Jan 03 2025

%F E.g.f.: exp(x)*(1 + 2*sin(x)). - _Stefano Spezia_, Jan 03 2025

%t Join[{a=1,b=3},Table[c=2*b-2*a+1;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 17 2011 *)

%t CoefficientList[Series[1/((1-2x+2x^2)(1-x)),{x,0,60}],x] (* or *) LinearRecurrence[{3,-4,2},{1,3,5},60] (* _Harvey P. Dale_, Feb 01 2013 *)

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

%o (PARI) a(n) = 1 + 2*imag((1 + I)^n); \\ _Daniel Suteu_, Dec 21 2018

%o (Maxima) a(n):=sum((-1)^k*2^(n-k)*binomial(n-k-1,k),k,0,n); /* _Vladimir Kruchinin_, Jul 02 2015 */

%o (Magma) I:=[1,3,5]; [n le 3 select I[n] else 3*Self(n-1)-4*Self(n-2)+2*Self(n-3): n in [1..60]]; // _Vincenzo Librandi_, Jul 02 2015

%K sign,easy,changed

%O 0,2

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