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 A137993 #7 Feb 01 2015 09:37:50
%S A137993 1,0,2,1,1,1,1,0,2,1,2,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,2,1,2,0,1,1,1,1,
%T A137993 2,0,1,0,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A137993 1,1,1,1,1,1,1,1,1,1,1,0
%N A137993 A014138 (= partial sums of Catalan numbers starting with 1,2,5) mod 3.
%C A137993 As usual, "mod 3" means to choose the unique representative in { 0,1,2 } of the equivalence class modulo 3Z.
%C A137993 Here the conventions of A014138 are used, but it seems somehow unnatural to start with offset 0 corresponding to the Catalan number A000108(1).
%C A137993 For m>1, the length of the m-th block of nonzero elements (and thus the approximate length of the m-th string of consecutive 1's) is given by 2 A137822(m)-1.
%F A137993 a(n) = sum( k=1..n+1, C(k) ) (mod 3), where C(k) = binomial(2k,k)/(k+1) = A000108(k).
%F A137993 a(n) = 0 <=> n+1 = 2 A137821(m) for some m.
%o A137993 (PARI) A137993(n) = lift( sum( k=1,n+1, binomial( 2*k,k )/(k+1), Mod(0,3) ))
%Y A137993 Cf. A014138, A000108, A137821-A137824, A107755, A137992, A014137(n+1) = a(n)+1 (mod 3).
%K A137993 easy,nonn
%O A137993 0,3
%A A137993 _M. F. Hasler_, Mar 16 2008