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a(n) = n-F(F(F(F(n)))) where F(x)=A120613(x)=floor(phi*floor(x/phi)) and phi=(1+sqrt(5))/2.
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%I #15 Sep 05 2024 08:10:07

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

%T 6,6,7,6,6,7,6,7,6,6,7,6,5,6,6,7,6,6,7,6,7,6,6,7,6,5,6,6,7,6,6,7,6,5,

%U 6,6,7,6,6,7,6,7,6,6,7,6,5,6,6,7,6,6,7,6,5,6,6,7,6,6,7,6,7,6,6,7,6,5,6,6,7

%N a(n) = n-F(F(F(F(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>=1 giving 0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,..... Then replace each 0 by the block {5,6,6} and each 1 by the block {7, 6, 6, 7, 6}. Append the initial string {1,2,3,4}.

%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(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), A110011 (k=5).

%K nonn,easy

%O 1,2

%A _Benoit Cloitre_, Sep 02 2005