A176866 The number of odd numbers that require n Collatz (3x+1) iterations to reach 1.

1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 4, 4, 6, 5, 7, 8, 14, 14, 19, 22, 30, 36, 48, 60, 79, 94, 118, 154, 194, 248, 315, 390, 486, 623, 792, 1008, 1261, 1579, 2007, 2555, 3219, 4043, 5109, 6464, 8204, 10351, 13100, 16575, 20889, 26398, 33388, 42155, 53370, 67414

Offset: 0

Author

T. D. Noe, Apr 27 2010

Keywords

• nonn

Comments

Both the 3x+1 steps and the halving steps are counted. The asymptotic growth rate appears to be the same as A005186, about 1.26 (A176014).

a(n) is, for n >= 4, the number of 4 (mod 6) nodes (vertices) of row n-1 of the Collatz tree A127824. The node 4 has in A127824 outdegree 1 in order to avoid a repetition of the whole tree. - Wolfdieter Lang, Mar 26 2014

The heuristic arguments given in the LINKS of A005186 suggest that this sequence has the same asymptotic growth rate (3+sqrt(21))/6. - Markus Sigg, Sep 07 2024

Examples

			23, 141, 151, 853, 909, and 5461 are the only odd numbers that require exactly 15 iterations to reach 1. Hence a(15)=6.
At row 15 with a(16) = 5 nodes 4 (mod 6) the left-right symmetry for the number of 4 (mod 6) nodes in the Collatz tree A127824 is broken for the first time: in the left half of the tree there are the three nodes 22, 136 and 832 but on the right half only the two nodes 904 and 5440. - _Wolfdieter Lang_, Mar 26 2014
		

Links

• Markus Sigg, Table of n, a(n) for n = 0..125 (first 71 terms from T. D. Noe).
• Wolfdieter Lang, On Collatz' Words, Sequences and Trees, arXiv preprint arXiv:1404.2710, 2014 and J. Int. Seq. 17 (2014) # 14.11.7.

Crossrefs

Cf. A005186 (number of numbers having stopping time n).

Cf. A127824 (numbers having stopping time n).

Cf. A176014, A176867, A176868.