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A176999
An encoding of the Collatz iteration of n.
3
1, 1111010, 11, 11110, 11110101, 1111011101101010, 111, 1111011101101010110, 111101, 11110111011010, 111101011, 111101110, 11110111011010101, 11110111110101010, 1111, 111101110110, 11110111011010101101, 11110111011010111010, 1111011, 1111110, 111101110110101
OFFSET
2,2
COMMENTS
Working from right to left, the sequence of 0's and 1's in a(n) encode, respectively, the sequence of 3x+1 and x/2 steps in the Collatz iteration of n. This is reverse one's complement of Garner's parity vector. Criswell mentions this encoding.
The length of a(n) is A006577(n). The number of 1's in a(n) is A006666(n). The number of 0's in a(n) is A006667(n). The number of terms having length k is A005186(k).
LINKS
Evans A. Criswell, The Collatz Problem (3x+1)
Lynn E. Garner, On heights in the Collatz 3n+1 problem, Discrete Math, 55 (1985), 57-64.
EXAMPLE
a(5)=11110 because the Collatz iteration for 5 is a 3x+1 step (0) followed by 4 x/2 steps (four 1's).
MATHEMATICA
encode[n_]:=Module[{m=n, p, lst={}}, While[m>1, p=Mod[m, 2]; AppendTo[lst, 1-p]; If[p==0, m=m/2, m=3m+1]]; FromDigits[Reverse[lst]]]; Table[encode[n], {n, 2, 26}]
CROSSREFS
KEYWORD
nonn,base
AUTHOR
T. D. Noe, Apr 30 2010
STATUS
approved