This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A280341 #24 Dec 16 2020 19:32:14 %S A280341 1,2,4,6,10,16,26,41,53,64,74,85,101,118,128,144,157,174,195,217,238, %T A280341 261,281,309,324,342,364,397 %N A280341 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 A280341 Here the height is defined to be the number of halving and tripling steps required to reach 1. %C A280341 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 A280341 Question: Is this sequence always increasing? %C A280341 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 A280341 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 A280341 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 A280341 collatz[n_] := If[EvenQ[n], n/2, 3n+1] %t A280341 height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1 %t A280341 a280341[n_] := Length[Union[Map[height, Range[2^n, 2^(n+1)-1]]]] %t A280341 (* sequence data; long computation times for n >= 22 *) %t A280341 Map[a280341, Range[0, 27]] %t A280341 (* _Hartmut F. W. Hoft_, Dec 16 2020 *) %Y A280341 Cf. A006577, A277109. %K A280341 nonn %O A280341 0,2 %A A280341 _Dmitry Kamenetsky_, Jan 01 2017 %E A280341 a(25)-a(27) from _Hartmut F. W. Hoft_, Dec 16 2020