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 A105730 #13 Jul 23 2024 01:01:15
%S A105730 6,12,8,6,13,8,6,9,11,6,21,8,6,78,8,6,9,13,6,15,8,6,16,8,6,9,20,6,12,
%T A105730 8,6,13,8,6,9,11,6,14,8,6,32,8,6,9,32,6,23,8,6,24,8,6,9,14,6,12,8,6,
%U A105730 13,8,6,9,11,6,14,8,6,19,8,6,9,13,6,80,8,6,29,8,6,9,18,6,12,8,6,13,8,6,9,11
%N A105730 Number of different initial values for 3x+1 trajectories in which the largest term appearing in the iteration is 2^(6n+4).
%C A105730 From _Hartmut F. W. Hoft_, Jun 24 2016: (Start)
%C A105730 The sequence has the quasiperiod 6, x, 8, 6, y, 8, 6, 9, z of length 9 starting at index 0 where x, y, z > 10; in addition, a(3*9*n+1) = 12, a(3*9*n+4) = 13 and a(3*9*n+8) = 11 for all n>=0; proof by induction (see this link) as in the link in A087256 based on the modular identities in the link in A033496.
%C A105730 Conjecture: All numbers greater than 10 appear in the sequence (see also A033496 and A233293). (End)
%H A105730 Hartmut F. W. Hoft, <a href="/A105730/a105730.pdf">identities to prove for quasi period 9</a>
%F A105730 a(n) = A087256(6n+4).
%e A105730 a(1) = 12, i.e. the number of initial values for 2^10, since 804 -> 402 -> 201 -> 604 -> 302 -> 151 -> 454 -> 227 -> 682 -> 341 -> 1024 and 908 -> (454 -> ... -> 1024) are the two maximal trajectories containing all 12 initial values. a(8) = 11 since 2^(6*8+4) has 11 different initial values for Collatz trajectories leading to it. - _Hartmut F. W. Hoft_, Jun 24 2016
%t A105730 trajectory[start_] := NestWhileList[If[OddQ[#], 3#+1, #/2]&, start, #!=1&]
%t A105730 fanSize[max_] := Module[{active={max}, fan={}, current}, While[active!={}, current=First[active];active=Rest[active]; AppendTo[fan, current]; If[2*current<=max, AppendTo[active, 2*current]]; If[Mod[current, 3]==1 && OddQ[(current-1)/3] && current>4, AppendTo[active, (current-1)/3]]]; Length[fan]]/;max==Max[trajectory[max]]
%t A105730 a105730[low_, high_] := Map[fanSize[2^(6#+4)]&, Range[low, high]]
%t A105730 a105730[0,89] (* _Hartmut F. W. Hoft_, Jun 24 2016 *)
%Y A105730 Cf. A025586, A033496, A087256, A233293.
%K A105730 easy,nonn,less
%O A105730 0,1
%A A105730 _David Wasserman_, Apr 18 2005