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 A351122 #81 May 12 2024 10:05:25
%S A351122 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,
%T A351122 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,
%U A351122 2,1,1,1,1,1,3,1,1,1,4,2,2,4,3,1,1,5,4,3,1,2,3,4
%N 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.
%F A351122 T(n,k) = log_2( (3*A256598(n,k)+1) / A256598(n,k+1) ).
%e A351122 Triangle starts at T(1,0):
%e A351122 n\k 0 1 2 3 4 5 6 7 8 ...
%e A351122 1: 1 4
%e A351122 2: 4
%e A351122 3: 1 1 2 3 4
%e A351122 4: 2 1 1 2 3 4
%e A351122 5: 1 2 3 4
%e A351122 6: 3 4
%e A351122 7: 1 1 1 5 4
%e A351122 8: 2 3 4
%e A351122 9: 1 3 1 2 3 4
%e A351122 10: 6
%e A351122 11: 1 1 5 4
%e A351122 12: 2 1 3 1 2 3 4
%e A351122 13: 1 2 1 1 1 1 2 2 1 2 1 1 2 ... (see A372362)
%e A351122 ...
%e A351122 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.
%e A351122 13 => 40 -> 20 -> 10 -> 5 => 16 -> 8 -> 4 -> 2 -> 1
%e A351122 triple \------------/ triple \---------------/
%e A351122 3 halvings 4 halvings
%e A351122 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
%e A351122 (((((((23*3+1)/2^1)*3+1)/2^1)*3+1)/2^5)*3+1)/2^4 = 1.
%o A351122 (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
%Y A351122 Cf. A075680 (row lengths), A166549 (row sums), A351123 (row partial sums).
%Y A351122 Cf. A256598.
%Y A351122 Cf. A020988 (where row is [2*n]).
%Y A351122 Cf. A198584 (where row length is 2), A228871 (where row is [1, x]).
%Y A351122 Cf. A372362 (row 13, the first 41 terms).
%K A351122 nonn,easy,tabf
%O A351122 1,2
%A A351122 _Flávio V. Fernandes_, Feb 01 2022
%E A351122 Corrected by _Michel Marcus_, Jul 18 2022