A176866 - OEIS

%I #27 Sep 07 2024 16:28:42

%S 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,

%T 118,154,194,248,315,390,486,623,792,1008,1261,1579,2007,2555,3219,

%U 4043,5109,6464,8204,10351,13100,16575,20889,26398,33388,42155,53370,67414

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

%C 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).

%C 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

%C 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

%H Markus Sigg, <a href="/A176866/b176866.txt">Table of n, a(n) for n = 0..125</a> (first 71 terms from T. D. Noe).

%H Wolfdieter Lang, <a href="http://arxiv.org/abs/1404.2710">On Collatz' Words, Sequences and Trees</a>, arXiv preprint arXiv:1404.2710, 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Lang/lang6.html">J. Int. Seq. 17 (2014) # 14.11.7</a>.

%e 23, 141, 151, 853, 909, and 5461 are the only odd numbers that require exactly 15 iterations to reach 1. Hence a(15)=6.

%e 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

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

%Y Cf. A127824 (numbers having stopping time n).

%Y Cf. A176014, A176867, A176868.

%K nonn

%O 0,8

%A _T. D. Noe_, Apr 27 2010