This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A110007 #19 Sep 04 2024 17:49:10
%S A110007 1,2,3,4,4,5,4,5,5,4,5,4,4,5,4,5,5,4,5,4,5,5,4,5,4,4,5,4,5,5,4,5,4,4,
%T A110007 5,4,5,5,4,5,4,5,5,4,5,4,4,5,4,5,5,4,5,4,5,5,4,5,4,4,5,4,5,5,4,5,4,4,
%U A110007 5,4,5,5,4,5,4,5,5,4,5,4,4,5,4,5,5,4,5,4,4,5,4,5,5,4,5,4,5,5,4,5,4,4,5,4,5
%N A110007 a(n) = n-F(F(F(n))) where F(x)=A120613(x)=floor(phi*floor(x/phi)) and phi=(1+sqrt(5))/2.
%C A110007 To build 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 {4,5,4} and each 1 by the block {5,5,4,5,4}. Append the initial string {1,2,3,4}.
%D A110007 Benoit Cloitre, On properties of irrational numbers related to the floor function, in preparation, 2005.
%t A110007 Join[{1,2,3,4},Flatten[Table[Floor[k/GoldenRatio]-Floor[(k-1)/ GoldenRatio],{k,30}]/.{0->{4,5,4},1->{5,5,4,5,4}}]] (* _Harvey P. Dale_, Dec 12 2017 *)
%o A110007 (PARI) F(x)=floor((1+sqrt(5))/2*floor((-1+sqrt(5))/2*x))
%o A110007 a(n)=n-F(F(F(n)))
%Y A110007 Cf. A005614 (infinite Fibonacci binary word), A120613.
%Y A110007 Cf. sequences for a(n) = n-F^k(n): A003842 (k=1), A110006 (k=2), A110010 (k=4), A110011 (k=5).
%K A110007 nonn,easy
%O A110007 1,2
%A A110007 _Benoit Cloitre_, Sep 02 2005