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a(0) = 0 and a(1) = 1, continued such that absolute values of 2nd differences equal the original sequence.
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%I #35 Apr 22 2023 07:13:37

%S 0,1,2,2,4,4,8,8,16,16,32,32,64,64,128,128,256,256,512,512,1024,1024,

%T 2048,2048,4096,4096,8192,8192,16384,16384,32768,32768,65536,65536,

%U 131072,131072,262144,262144,524288,524288,1048576,1048576

%N a(0) = 0 and a(1) = 1, continued such that absolute values of 2nd differences equal the original sequence.

%C This is the main sequence of a family of sequences starting at a(0) = A and a(1) = B, continuing a(3, ...) = 2B, 2B, 4B, 4B, 8B, 8B, 16B, 16B, 32B, 32B, ... such that the absolute values of the 2nd differences, abs(a(n+2) - 2*a(n+1) + a(n)), equal the original sequence. Alternatively starting at a(0) = a(1) = 1 gives A016116.

%H Vincenzo Librandi, <a href="/A131572/b131572.txt">Table of n, a(n) for n = 0..2000</a>

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

%F a(n) = 2*a(n-2), n>2.

%F O.g.f.: x*(1+2*x)/(1-2*x^2). - _R. J. Mathar_, Jul 16 2008

%F a(n) = A016116(n) - A000007(n), that is, a(0)=0, a(n) = A016116(n) for n>=1. - _Bruno Berselli_, Apr 13 2011

%F First differences: a(n+1) - a(n) = A131575(n).

%F Second differences: A131575(n+1) - A131575(n) = (-1)^n*a(n).

%F E.g.f.: -1 + cosh(sqrt(2)*x) + (1/sqrt(2))*sinh(sqrt(2)*x). - _G. C. Greubel_, Apr 22 2023

%t LinearRecurrence[{0,2},{0,1,2},50] (* _Harvey P. Dale_, Jul 10 2018 *)

%o (Magma) [2^Floor(n/2)-0^n: n in [0..50]]; // _Vincenzo Librandi_, Aug 18 2011

%o (SageMath) [0]+[2^(n//2) for n in range(1,51)] # _G. C. Greubel_, Apr 22 2023

%Y Cf. A000007, A131575.

%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 nonn,easy

%O 0,3

%A _Paul Curtz_, Aug 28 2007

%E Edited by _R. J. Mathar_, Jul 16 2008

%E More terms from _Vincenzo Librandi_, Aug 18 2011