A351122 Irregular triangle read by rows in which row n lists the number of divisions by 2 after tripling steps in the Collatz 3x+1 trajectory of 2n+1 until it reaches 1.

1, 4, 4, 1, 1, 2, 3, 4, 2, 1, 1, 2, 3, 4, 1, 2, 3, 4, 3, 4, 1, 1, 1, 5, 4, 2, 3, 4, 1, 3, 1, 2, 3, 4, 6, 1, 1, 5, 4, 2, 1, 3, 1, 2, 3, 4, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 2, 2, 4, 3, 1, 1, 5, 4, 3, 1, 2, 3, 4

Offset

1,2

Links

Table of n, a(n) for n=1..92.

Formula

T(n,k) = log_2( (3*A256598(n,k)+1) / A256598(n,k+1) ).

Example

Triangle starts at T(1,0):
   n\k   0   1   2   3   4   5   6   7   8 ...
   1:    1   4
   2:    4
   3:    1   1   2   3   4
   4:    2   1   1   2   3   4
   5:    1   2   3   4
   6:    3   4
   7:    1   1   1   5   4
   8:    2   3   4
   9:    1   3   1   2   3   4
  10:    6
  11:    1   1   5   4
  12:    2   1   3   1   2   3   4
  13:    1   2   1   1   1   1   2   2   1   2   1   1   2  ... (see A372362)
  ...
For n=6, the trajectory of 2*n+1 = 13 is as follows. The tripling steps ("=>") are followed by runs of 3 and then 4 halvings ("->"), so row n=6 is 3, 4.
  13  =>  40 -> 20 -> 10 -> 5  =>  16 -> 8 -> 4 -> 2 -> 1
    triple   \------------/   triple  \---------------/
               3 halvings                4 halvings
Runs of halvings are divisions by 2^T(n,k). Row n=11 is 1, 1, 5, 4 and its steps starting from 2*n+1 = 23 reach 1 by a nested expression
  (((((((23*3+1)/2^1)*3+1)/2^1)*3+1)/2^5)*3+1)/2^4 = 1.

Prog

(PARI) row(n) = my(m=2*n+1, list=List()); while (m != 1, if (m%2, m = 3*m+1, my(nb = valuation(m, 2)); m/=2^nb; listput(list, nb)); ); Vec(list); \\ Michel Marcus, Jul 18 2022

Crossrefs

Cf. A075680 (row lengths), A166549 (row sums), A351123 (row partial sums).

Cf. A256598.

Cf. A020988 (where row is [2*n]).

Cf. A198584 (where row length is 2), A228871 (where row is [1, x]).

Cf. A372362 (row 13, the first 41 terms).

Sequence in context: A143484 A047214 A307886 * A214499 A234002 A016496

Adjacent sequences: A351119 A351120 A351121 * A351123 A351124 A351125

Keyword

nonn,easy,tabf

Author

Flávio V. Fernandes, Feb 01 2022

Extensions

Corrected by Michel Marcus, Jul 18 2022

Status

approved