A Collatz-like sequence

[Nick Hobson]

Hi Seqfans,

Is this Collatz-like sequence of interest?  a(0) = 0; for n > 0, a(n) =  
a(n-1)/2 if a(n-1) is even and not already in the sequence, otherwise a(n)  
= 3*a(n-1) + 1.  The first few terms...

0, 1, 4, 2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 25, 76,  
38, 19, 58, 29, 88, 44, 133, 400, 200, 100, 50, 151, 454, 227, 682, 341,  
1024, 512, 256, 128, 64, 32, 97, 292, 146, 73, 220, 110, 55, 166, 83, 250,  
125, 376, 188, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364,  
182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526,  
263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251,  
754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719,  
2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, ...

It's easy to see that, apart from 0, the sequence misses multiples of 3.   
The sequence below shows the index where n first appears, or None if  
never.  The first non-multiple of 3 that is not present in the first 10  
million terms is 43925.  Does the sequence a(n) eventually hit every  
non-multiple of 3?

0, 1, 3, None, 2, 15, None, 4, 17, None, 14, 6, None, 11, 182, None, 16,  
8, None, 21, 13, None, 5, 145, None, 18, 10, None, 181, 23, None, 349, 41,  
None, 7, 147, None, 178, 20, None, 12, 346, None, 183, 25, None, 144, 56,  
None, 188, 30, None, 9, 149, None, 48, 180, None, 22, 317, None, 141, 348,  
None, 40, 185, None, 322, 159, None, 146, 58, None, 45, 177, None, 19,  
164, None, 3450, 151, None, 345, 50, None, 738, 588, None, 24, 319, None,  
68, 143, None, 55, 637, None, 42, 187, None, ... .

The idea of a "memory" was inspired by Recaman's sequence (A005132): a(0)  
= 0; for n > 0, a(n) = a(n-1) - n if that number is positive and not  
already in the sequence, otherwise a(n) = a(n-1) + n.

Nick