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%I #28 Apr 28 2019 20:09:11
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,
%U 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N a(n) = x(y(n)) - y(x(n)) where x = A302128 and y = A005350.
%C A chaotic sequence based on a definition by A. Fraenkel. Fibonacci numbers determine the boundaries of the generations.
%H Robert Israel, <a href="/A307721/b307721.txt">Table of n, a(n) for n = 1..10000</a>
%H Altug Alkan, <a href="/A307721/a307721.png">Line plot of a(n) for n <= 28657</a>
%H A. S. Fraenkel, <a href="https://doi.org/10.1090/S0002-9947-1994-1138949-9">Iterated floor function, algebraic numbers, discrete chaos, Beatty subsequences, semigroups</a>, Trans. Amer. Math. Soc., Volume 341, Number 2 (1994).
%H Rémy Sigrist, <a href="/A307721/a307721.txt">C program for A307721</a>
%p x:= proc(n) option remember;
%p procname(procname(n-2)) + procname(n-procname(n-1))
%p end proc:
%p x(1):= 1: x(2):= 1: x(3):= 1:
%p y:= proc(n) option remember;
%p procname(procname(n-1)) + procname(n-procname(n-1))
%p end proc:
%p y(1):= 1: y(2):= 1: y(3):= 1:
%p map(x@y-y@x, [$1..100]); # _Robert Israel_, Apr 25 2019
%t x[1]=x[2]=x[3]=y[1]=y[2]=y[3]=1; x[n_] := x[n] = x[x[n-2]] + x[n - x[n - 1]]; y[n_] := y[n] = y[y[n-1]] + y[n - y[n-1]]; a[n_] := x[y[n]] - y[x[n]]; Array[a, 100] (* _Giovanni Resta_, Apr 24 2019 *)
%o (PARI) x=vector(200); for(n=1, 3, x[n] = 1); for(n=4, #x, x[n] = x[x[n-2]] + x[n-x[n-1]]); y=vector(200); for(n=1, 3, y[n] = 1); for(n=4, #y, y[n] = y[y[n-1]] + y[n-y[n-1]]); vector(200, n, x[y[n]]-y[x[n]])
%o (C) See Links section.
%Y Cf. A005350, A302128.
%K sign
%O 1
%A _Altug Alkan_, Apr 24 2019
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