Seqs increasing fast then slow then fast then...

[Leroy Quet]

I have recently submitted these sequences:

>%S A000001 1,2,4,5,9,14,15,16,17,26,27,28,29,30,44,59,75,92,93,94,95
>%N A000001 a(0)=1. a(n) = a(n-1) + n, if n is in the sequence. a(n) = 
>a(n-1) + 1 if n is missing from the sequence.
>%O A000001 0
>%K A000001 ,easy,more,nonn,

>%S A000001 
>1,2,4,5,10,20,21,22,23,24,48,49,50,51,52,53,54,55,56,57,114,228,456,912,18
>24,1825,1826,1827,1828,1829,1830,1831,1832,1833,1834,1835
>%N A000001 a(0)=1. a(n) = a(n-1)*2, if n is in the sequence. a(n) = a(n-1) 
>+ 1 if n is missing from the sequence.
>%O A000001 0
>%K A000001 ,easy,nonn,

>%S A000001 
>1,2,6,7,8,9,63,504,4536,45360,45361,45362,45363,45364,45365,45366,45367,45
>368,45369,45370,45371,45372,45373,45374,45375,45376,45377,45378,45379,4538
>0,45381,45382,45383,45384,45385,45386,45387,45388,45389,45390
>%N A000001 a(0)=1. a(n) = a(n-1)*(n+1), if n is in the sequence. a(n)= 
>a(n-1) +1, if n is missing from the sequence.
>%O A000001 0
>%K A000001 ,easy,nonn,

These sequences are unusual in that they alternate increasing rapidly and 
slowly (by "increasing slowly", I mean icreasing by 1 each term), and do 
so in a manner dictated by earlier terms of the same sequence.  (The 2nd 
sequence increases more rapidly than the 1st, the 3rd more rapidly than 
the 2nd, of course.)

How would someone go about finding a function which approaches the terms 
of such sequences asymptotically?
And what is a closed-form explicit formula for the terms of these 
sequences? (I am betting the first sequence's formula is actually easy to 
discover, the other sequences formula's are much harder, if at all 
possible. Just a guess; I haven't tried to find closed forms for any of 
these sequences myself yet.)

thanks,
Leroy Quet