A099430 - OEIS

%I #16 Jan 08 2025 10:11:13

%S 1,0,4,6,16,30,64,126,256,510,1024,2046,4096,8190,16384,32766,65536,

%T 131070,262144,524286,1048576,2097150,4194304,8388606,16777216,

%U 33554430,67108864,134217726,268435456,536870910,1073741824,2147483646

%N a(n) = 2^n+(-1)^n-1.

%C Jacobsthal-Lucas numbers less 1.

%C For n >= 1, a(n) is also the number of periodic points of period n of the Period Doubling and the Thue-Morse Chain. [Natascha Neumaerker (naneumae(AT)math.uni-bielefeld.de), Apr 06 2009]

%H N. Neumarker, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Neumarker/neumarker3.html">Realizability of Integer Sequences as Differences of Fixed Point Count Sequences</a>, JIS 12 (2009) 09.4.5, Example 9.

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

%F G.f.: (1-2x+3x^2)/((1-x)(1-x-2x^2)) = (1-2x+3x^2)/((1-x^2)(1-2x)).

%F a(n) = A014551(n)-1.

%F zeta(z) = (1-z)/((1+z)(1-2z)). [Natascha Neumaerker (naneumae(AT)math.uni-bielefeld.de), Apr 06 2009]

%F a(n) = 2*a(n-1)+a(n-2)-2*a(n-3). - _Wesley Ivan Hurt_, Jun 09 2023

%t LinearRecurrence[{2,1,-2},{1,0,4},40] (* _Harvey P. Dale_, Nov 07 2017 *)

%o (PARI) a(n)=2^n+(-1)^n-1 \\ _Charles R Greathouse IV_, May 09 2016

%Y Cf. A014551.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Oct 15 2004