%N Number of distinct heights achieved in the Collatz (or '3x+1') problem when starting from numbers in the range [2^n,2^(n+1)).
%C Here the height is defined to be the number of halving and tripling steps required to reach 1.
%C Interestingly the values in this sequence grow slowly (almost linearly) indicating that the average number of starting values with the same height increases with n.
%C Question: Is this sequence always increasing?
%C Definition corrected by _N. J. A. Sloane_, Apr 09 2020. The old definition was "Number of unique heights achieved in the Collatz (or '3x+1') problem when starting from numbers in the range [2^n,2^(n+1))."
%C This sequence a(n) as well as the sequence of maximum heights in each interval appear to increase quadratically with n. The odd numbers in [2^n, 2^(n+1)) , 5 <= n <= 20, create all distinct heights for the interval except for height n of number 2^n, and except for height n+3 when n is odd. - _Hartmut F. W. Hoft_, Dec 16 2020
%e The heights for starting values 16 to 31 are: 4, 12, 20, 20, 7, 7, 15, 15, 10, 23, 10, 111, 18, 18, 18, 106. The unique heights are: 4, 12, 20, 7, 15, 10, 23, 111, 18, 106. Hence a(4)=10.
%t collatz[n_] := If[EvenQ[n], n/2, 3n+1]
%t height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1
%t a280341[n_] := Length[Union[Map[height, Range[2^n, 2^(n+1)-1]]]]
%t (* sequence data; long computation times for n >= 22 *)
%t Map[a280341, Range[0, 27]]
%t (* _Hartmut F. W. Hoft_, Dec 16 2020 *)
%Y Cf. A006577, A277109.
%A _Dmitry Kamenetsky_, Jan 01 2017
%E a(25)-a(27) from _Hartmut F. W. Hoft_, Dec 16 2020