

A233135


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


4



1, 2, 21, 22, 221, 212, 2121, 222, 2221, 2212, 22121, 2122, 21221, 21212, 212121, 2222, 22221, 22212, 222121, 22122, 221221, 221212, 2212121, 21222, 212221, 212212, 2122121, 212122, 2121221, 2121212, 21212121, 22222, 222221, 222212, 2222121, 222122, 2221221
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OFFSET

1,2


COMMENTS

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(n1), 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.


LINKS



FORMULA

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,k1)). 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.


MATHEMATICA

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 *)
Table[FromDigits[Reverse[u[n]]], {n, 1, 30}] (* A233135 *)
Flatten[Table[Reverse[u[n]], {n, 1, 30}]] (* A233136 *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



