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 A188590 #9 Nov 21 2013 13:12:28
%S A188590 3,3,4,3,3,4,3,3,4,3,3,3,4,3,3,4,3,3,4,3,3,3,4,3,3,4,3,3,4,3,3,3,4,3,
%T A188590 3,4,3,3,4,3,3,4,3,3,3,4,3,3,4,3,3,4,3,3,3,4,3,3,4,3,3,4,3,3,3,4,3,3,
%U A188590 4,3,3,4,3,3,4,3
%N A188590 [(n+1)*r] - [n*r], where r = 3/2 + sqrt(13)/2 and [...] denotes the floor function.
%C A188590 It appears that this sequence is a fixed-pt of the morphism 3 -> 334, 4 -> 3343, starting with 3. The orbit of 3 under the indicated morphism is 3, 334, 3343343343, 334334334333433433433343343343334, ...
%C A188590 The sequence of the lengths of the words in this orbit appears to be A006190 = {1,3,10,33,109,...}, a solution of the difference equation a(n) = 3*a(n-1) + a(n-2). A root of the auxiliary equation r^2 - 3r -1 = 0 of this difference equation is 3/2 + sqrt(13)/2, the value of r used in the definition of {a(n)}.
%C A188590 See A003849 for the infinite Fibonacci word (start with 0, apply 0->01, 1->0, take limit).
%C A188590 It appears that {a(n)-1} = {2,2,3,2,2,3,2,2,3,2,2,2,3,...} is the same as A003589 (the number of 2's between consecutive 3's in A003589 gives the original sequence). This has been verified up to 2000 terms.
%F A188590 a(n) = [(n+1)*r] - [n*r], where r = 3/2 + sqrt(13)/2 and [...] denotes the floor function.
%t A188590 r = 3/2 + Sqrt[13]/2; Table[Floor[(n + 1)r] - Floor[n * r], {n, 100}] (* _Alonso del Arte_, Apr 04 2011 *)
%Y A188590 Cf. A003589, A003849, A006190.
%K A188590 nonn
%O A188590 1,1
%A A188590 _John W. Layman_, Apr 04 2011