A365819 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.

0, 1, 0, -2, -3, -4, -5, -7, -8, -9, -10, -11, -12, -13, -15, -17, -19, -21, -22, -23, -24, -25, -26, -27, -27, -27, -27, -27, -28, -29, -31, -33, -36, -39, -43, -47, -51, -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

Offset: 0

Author

Andres Cicuttin, Dec 14 2023

Keywords

• sign

Comments

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

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

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Thomas Scheuerle, Plot a(0)..a(20000).
• Thomas Scheuerle, Plot n = 0..200000, 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.

Programs

MATLAB

function a = A365819( max_n )
    a = [0 1];
    for n = 3:max_n
        a(n) = a(n-1) - 1 + floor(median(a) - mean(a));
    end
end % Thomas Scheuerle, Dec 15 2023

Maple

R:= [0,1]:
for i from 2 to 10000 do
  v:= R[-1] - 1 + floor(Statistics:-Median(R) - Statistics:-Mean(R));
  R:= [op(R),v]
od:
R; # Robert Israel, Oct 14 2024

Mathematica

a = {0, 1};
Do[AppendTo[a, Last[a] - 1 + Floor[Median[a] - Mean[a]]], {j, 1, 100}]
a[[1 ;; 100]]