A143096 - OEIS

%I #19 Jul 24 2023 10:08:01

%S 1,4,5,9,17,33,65,129,257,513,1025,2049,4097,8193,16385,32769,65537,

%T 131073,262145,524289,1048577,2097153,4194305,8388609,16777217,

%U 33554433,67108865,134217729,268435457,536870913,1073741825,2147483649,4294967297,8589934593

%N a(n) = 2*a(n-1)-1, with a(1)=1, a(2)=4, a(3)=5.

%H Vincenzo Librandi, <a href="/A143096/b143096.txt">Table of n, a(n) for n = 1..2000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).

%F Binomial transform of 0, 1, 2, -4, 9, -13, 20, -26, 35, ... (offset 0).

%F O.g.f.: x*(1+x-5*x^2+2*x^3)/((1-x)*(1-2*x)). a(n) = 1+2^(n-1), n>2. - R. J. Mathar, Jul 31 2008

%F a(n) = A048578(n-2), n>=3. - R. J. Mathar, Aug 10 2008

%e a(4) = 9 = 2*a(3) - 1 = 2*5 - 1.

%e a(4) = 9 = (1, 3, 3, 1) dot (1, 3, -2, 5) = (1 + 9 - 6 + 5).

%t Join[{1,4},NestList[2#-1&,5,40]] (* or *) Join[{1,4},LinearRecurrence[ {3,-2},{5,9},40]] (* _Harvey P. Dale_, Feb 18 2014 *)

%Y Cf. A065190.

%Y Essentially the same as A083318, A048578 and A000051.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_ & _Roger L. Bagula_, Jul 23 2008