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.

1, 2, 3, 4, 10, 13, 33, 55, 112, 181, 352, 580, 1072, 2127, 6792, 13067, 25906, 51447, 104575, 208149, 415921, 833109, 1661341, 3328124, 6648354, 13283680, 26533708, 53083687, 106166631, 212243709, 424564626, 848967377, 1698139390, 3396064464, 6791623786

Offset: 0

Author

Labos Elemer, Jun 14 2004

Keywords

• nonn

Examples

			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
		

Links

• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A087256, A095381, A095382, A095383.

Cf. A006884, A006885, A222292, A224538, A224540.

Programs

Maple

b:= proc(n) option remember; `if`(n=1, 1,
      max(n, b(`if`(n::even, n/2, 3*n+1))))
    end:
a:= proc(n) option remember; local t; t:=2^n;
      add(`if`(b(i)<=t, 1, 0), i=1..t)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 26 2024

Mathematica

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}]
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 *)

Extensions

a(21)-a(32) from Donovan Johnson, Feb 02 2011

a(0) from T. D. Noe, Apr 29 2013

a(33)-a(34) from Donovan Johnson, Jun 05 2013