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Even bisection of the infinite Fibonacci word A003849.
5

%I #10 Feb 27 2021 21:10:35

%S 0,0,1,1,0,0,1,1,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,1,1,0,0,0,1,0,0,0,1,0,

%T 0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0,

%U 0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0,0

%N Even bisection of the infinite Fibonacci word A003849.

%F a(n) = 2 - [(2n+2)r] + [(2n+1)r], where [ ] = floor and r = golden ratio (A001622).

%e A003849 = (0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, ...), so that

%e A339824 = (0, 0, 1, 1, 0, 0, 1, ...), the even bisection, and

%e A339825 = (1, 0, 0, 0, 1, 0, 0, ...), the odd bisection.

%t r = (1 + Sqrt[5])/2; z = 300;

%t f[n_] := 2 - Floor[(n + 2) r] + Floor[(n + 1) r]; (* A003849 *)

%t Table[2 - Floor[(2 n + 2) r] + Floor[(2 n + 1) r], {n, 0, Floor[z/2]}](* A339824 *)

%t Table[2 - Floor[(2 n + 3) r] + Floor[(2 n + 2) r], {n, 0, Floor[z/2]}](* A339825 *)

%Y Cf. A001622, A096270, A339825, A339826, A339827.

%K nonn

%O 0

%A _Clark Kimberling_, Dec 19 2020

%E Corrected by _Michel Dekking_, Feb 23 2021