A152166 - OEIS

%I #26 Jul 19 2024 19:06:23

%S 1,-2,2,-4,4,-8,8,-16,16,-32,32,-64,64,-128,128,-256,256,-512,512,

%T -1024,1024,-2048,2048,-4096,4096,-8192,8192,-16384,16384,-32768,

%U 32768,-65536,65536,-131072,131072,-262144,262144,-524288,524288,-1048576,1048576

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

%C Ratios of successive terms are -2,-1,-2,-1,-2,-1,-2,-1,... - _Philippe Deléham_, Dec 12 2008

%H Paolo Xausa, <a href="/A152166/b152166.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,2).

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

%F a(n) = 2*a(n-2); a(0)=1, a(1)=-2.

%F a(n) = Sum_{k=0..n} A147703(n,k)*(-3)^k.

%F E.g.f.: cosh(sqrt(2)*x) - sqrt(2)*sinh(sqrt(2)*x). - _Stefano Spezia_, Feb 05 2023

%t LinearRecurrence[{0, 2}, {1, -2}, 50] (* _Paolo Xausa_, Jul 19 2024 *)

%Y Cf. A000079, A016116, A147703.

%Y The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - _N. J. A. Sloane_, Jul 14 2022

%K sign,easy

%O 0,2

%A _Philippe Deléham_, Nov 27 2008