A365819 - OEIS

%I #29 Oct 14 2024 23:57:43

%S 0,1,0,-2,-3,-4,-5,-7,-8,-9,-10,-11,-12,-13,-15,-17,-19,-21,-22,-23,

%T -24,-25,-26,-27,-27,-27,-27,-27,-28,-29,-31,-33,-36,-39,-43,-47,-51,

%U -54,-57,-59,-61,-63,-64,-65,-65,-65,-65,-64,-63,-61,-58,-55,-51,-47,-43,-39,-35,-31,-28,-25,-22,-19

%N a(n) = a(n-1) - 1 + floor(median(a) - mean(a)) where a(0) = 0, a(1) = 1, and median(a) and mean(a) are respectively the median and mean of all previous terms.

%C The median of an even number of terms is taken as the mean of its middle two values, (x+y)/2.

%C The sequence seems to present a chaotic pattern. What is its asymptotic behavior?

%H Robert Israel, <a href="/A365819/b365819.txt">Table of n, a(n) for n = 0..10000</a>

%H Thomas Scheuerle, <a href="/A365819/a365819.png">Plot a(0)..a(20000)</a>.

%H Thomas Scheuerle, <a href="/A365819/a365819_1.png">Plot n = 0..200000</a>, blue: a(n)/10, red: mean(a(0)..a(n)), orange: 10*(median(a(0)..a(n))-mean(a(0)..a(n))). We observe new emergent chaos after periods of apparent calming.

%p R:= [0,1]:

%p for i from 2 to 10000 do

%p   v:= R[-1] - 1 + floor(Statistics:-Median(R) - Statistics:-Mean(R));

%p   R:= [op(R),v]

%p od:

%p R; # _Robert Israel_, Oct 14 2024

%t a = {0, 1};

%t Do[AppendTo[a, Last[a] - 1 + Floor[Median[a] - Mean[a]]], {j, 1, 100}]

%t a[[1 ;; 100]]

%o (MATLAB)

%o function a = A365819( max_n )

%o     a = [0 1];

%o     for n = 3:max_n

%o         a(n) = a(n-1) - 1 + floor(median(a) - mean(a));

%o     end

%o end % _Thomas Scheuerle_, Dec 15 2023

%K sign

%O 0,4

%A _Andres Cicuttin_, Dec 14 2023