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A095384 Number of different initial values for 3x+1 trajectories started with initial values not exceeding 2^n and in which the peak values are also not larger than 2^n.

%I A095384 #23 Sep 26 2024 04:37:45
%S A095384 1,2,3,4,10,13,33,55,112,181,352,580,1072,2127,6792,13067,25906,51447,
%T A095384 104575,208149,415921,833109,1661341,3328124,6648354,13283680,
%U A095384 26533708,53083687,106166631,212243709,424564626,848967377,1698139390,3396064464,6791623786
%N A095384 Number of different initial values for 3x+1 trajectories started with initial values not exceeding 2^n and in which the peak values are also not larger than 2^n.
%H A095384 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A095384 n=4: between iv={1,2,...,16} {2,8}U{3,5,6,10,12,16} provides peak values smaller than or equal with 16, so a(4) = 10 = A087256(4)+4
%p A095384 b:= proc(n) option remember; `if`(n=1, 1,
%p A095384       max(n, b(`if`(n::even, n/2, 3*n+1))))
%p A095384     end:
%p A095384 a:= proc(n) option remember; local t; t:=2^n;
%p A095384       add(`if`(b(i)<=t, 1, 0), i=1..t)
%p A095384     end:
%p A095384 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 26 2024
%t A095384 c[x_]:=c[x]=(1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1);c[1]=1; fpl[x_]:=FixedPointList[c, x]; {$RecursionLimit=1000;m=0}; Table[Print[{xm-1, m}];m=0; Do[If[ !Greater[Max[fpl[n]], 2^xm], m=m+1], {n, 1, 2^xm}], {xm, 1, 30}]
%t A095384 Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Length[Select[Range[x=2^n], Max[Collatz[#]] <= x &]], {n,0,10}] (* _T. D. Noe_, Apr 29 2013 *)
%Y A095384 Cf. A087256, A095381, A095382, A095383.
%Y A095384 Cf. A006884, A006885, A222292, A224538, A224540.
%K A095384 nonn
%O A095384 0,2
%A A095384 _Labos Elemer_, Jun 14 2004
%E A095384 a(21)-a(32) from _Donovan Johnson_, Feb 02 2011
%E A095384 a(0) from _T. D. Noe_, Apr 29 2013
%E A095384 a(33)-a(34) from _Donovan Johnson_, Jun 05 2013