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A233135 Shortest (x+1,2x)-code of n.

%I A233135 #5 Dec 07 2013 12:56:18
%S A233135 1,2,21,22,221,212,2121,222,2221,2212,22121,2122,21221,21212,212121,
%T A233135 2222,22221,22212,222121,22122,221221,221212,2212121,21222,212221,
%U A233135 212212,2122121,212122,2121221,2121212,21212121,22222,222221,222212,2222121,222122,2221221
%N A233135 Shortest (x+1,2x)-code of n.
%C A233135 Every positive integer is a composite of f(x) = x + 1 and g(x) = 2*x starting with x = 1.  For example, 5 = f(g(g(1))), which abbreviates as fgg, or 122, which we call a (x+1,2x)-code of 5.  It appears that the number of (x+1,2x)-codes of n is A040039(n), that these numbers form Guy Steele's sequence GS(4,5) at A135529, and that for k >= 1, then number of such codes is F(n-1), where F = A000045, the Fibonacci numbers.  See A232559 for the uncoded positive integers in the order generated by the rules x -> x+1 and x -> 2*x.
%H A233135 Clark Kimberling, <a href="/A233135/b233135.txt">Table of n, a(n) for n = 1..1000</a>
%F A233135 Define h(x) = x - 1 if x is odd and h(x) = x/2 if x is even, and define H(x,1) = h(x) and H(x,k) = H(H(x,k-1)).  For each n > 1, the sequence (H(n,k)) decreases to 1 through two kinds of steps; write 1 when the step is x - 1 and write 2 when the step is x/2.  Let c(n) be the concatenation of 1s and 2s; then A233135(n) is the reversal of c(n), as in the Mathematica program.
%t A233135 b[x_] := b[x] = If[OddQ[x], x - 1, x/2]; u[n_] := 2 - Mod[Drop[FixedPointList[b, n], -3], 2]; u[1] = {1}; t = Table[u[n], {n, 1, 30}]; Table[FromDigits[u[n]], {n, 1, 50}]  (* A233137 *)
%t A233135 Flatten[t]  (* A233138 *)
%t A233135 Table[FromDigits[Reverse[u[n]]], {n, 1, 30}]  (* A233135 *)
%t A233135 Flatten[Table[Reverse[u[n]], {n, 1, 30}]]  (* A233136 *)
%Y A233135 Cf. A040039, A135529, A232559, A000045, A233136, A233137, A233138.
%K A233135 nonn,easy
%O A233135 1,2
%A A233135 _Clark Kimberling_, Dec 05 2013