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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.
2
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
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
KEYWORD
nonn,easy,tabf
AUTHOR
EXTENSIONS
Corrected by Michel Marcus, Jul 18 2022
STATUS
approved