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A233135
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Shortest (x+1,2x)-code of n.
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4
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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
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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