[seqfan] Three Jumping Sequences

[Neal Tolunsky]

Dear SeqFans,
My name is Neal (https://oeis.org/wiki/User:Neal_Gersh_Tolunsky). I'd like to ask about several related sequences not yet in the OEIS that are based on the concept of jumping: From every term a(i) you can jump a(i) terms in either direction to reach another term. 
So here's one sequence (definition would need work but you can get the idea): a(n+1) is the maximum number of connected terms in the sequence thus far, given any starting term, according to the rule that any term k can reach another term by jumping k terms in either direction; a(1)=1.
Example: a(5)=4 because the sequence so far looks like 1,1,2,3 and you can visit the greatest number of terms (4) by starting at 3, jumping to 1a, then forward to 1b, and finally to 2. By the way, you can visit the same term more than once.
This sequence begins: 1,1,2,3,4,5,5,6,6,7,7,9,10...
Another sequence is the least number of terms that are connected given any starting point (in this case least-connected is the same as number of connected terms starting at a(1)=1).
Begins: 1,1,2,3,3,4,4,4,7,7,7,8,9...
One more sequence would be: maximum number of terms you can reach starting at the most recent term, which is the most chaotic sequence of the three since it isn't bounded by the previous term. It starts: 1,1,2,3,4,4,5,5,5,7,8,8,8,9,9,12,10
As a non-mathematician, I'd just like to ask if anyone can find out anything about these sequences (if they're of any interest); how do they behave (why the jumps in the graph)? 
Thank you, 
Neal
P.S. I just noticed that S. Brunner has drafts for three similar sequences (A360593-5), which is encouraging. I plan to submit these once my other sequences are reviewed.