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a(n) = n-F(F(F(F(F(n))))) = n-F^5(n) where F(x)=A120613(x)=floor(phi*floor(x/phi)) and phi=(1+sqrt(5))/2.
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%I #12 Sep 04 2024 17:48:40

%S 1,2,3,4,5,6,7,8,8,7,8,8,7,8,7,8,8,7,8,8,9,8,7,8,8,7,8,7,8,8,7,8,8,7,

%T 8,7,8,8,7,8,8,9,8,7,8,8,7,8,7,8,8,7,8,8,9,8,7,8,8,7,8,7,8,8,7,8,8,7,

%U 8,7,8,8,7,8,8,9,8,7,8,8,7,8,7,8,8,7,8,8,7,8,7,8,8,7,8,8,9,8,7,8,8,7,8,7,8

%N a(n) = n-F(F(F(F(F(n))))) = n-F^5(n) where F(x)=A120613(x)=floor(phi*floor(x/phi)) and phi=(1+sqrt(5))/2.

%C To built the sequence start from the infinite Fibonacci word b(k)=floor(k/phi)-floor((k-1)/phi) for k>=2 giving 1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,..... Then replace each 0 by the block {9,8,7,8,8,7,8,7,8,8,7,8,8} and each 1 by the block {9,8,7,8,8,7,8,7,8,8,7,8,8,7,8,7,8,8,7,8,8}. Append the initial string {1,2,3,4,5,6,7,8,8,7,8,8,7,8,7,8,8,7,8,8}.

%D Benoit Cloitre, On properties of irrational numbers related to the floor function, in preparation, 2005.

%o (PARI) F(x)=floor((1+sqrt(5))/2*floor((-1+sqrt(5))/2*x)); a(n)=n-F(F(F(F(F(n)))))

%Y Cf. A005614 (infinite Fibonacci binary word), A120613.

%Y Cf. sequences for a(n) = n-F^k(n): A003842 (k=1), A110006 (k=2), A110007 (k=3), A110010 (k=4).

%K nonn,easy

%O 1,2

%A _Benoit Cloitre_, Sep 02 2005