Agreeing Until Disagreeing

[Leroy Quet]

I was debating whether I should send in this question to seq.fan. But 
since my question reminds me of Hans Havermann's recent question for 
seq.fan, I decided to send it in.

I recently submitted the following:

>%S A000001 0,2,4,7,10,14,19,24,30,37,44,52,61,71,81,92,104,117
>%N A000001 a(0)=0. a(n) = a(n-1) + 1 + (number of positive integers 
>missing from {a(0),a(1),a(2),...a(n-1)}).
>%e A000001 The sequence of positive integers not in this sequence begins 
>1, 3, 5, 6, 8,...
>Because there are four such terms <= 7, then a(7) = a(6) + 1 + 4 = 24.
>%O A000001 0
>%K A000001 ,easy,more,nonn,

Except for the initial 0, the sequence matches a couple others for most 
of the terms I have submitted.
Specifically, David Wilson's sequence A025711 matches my sequence for the 
terms I give except that the first term is 0 for mine, 1 for A025711, and 
the last term I give is 117, while this term is 116 for David Wilson's 
sequence.

As Hans had asked in his question: is there a number-theoretical reason 
for the closeness of the sequences?

(Also kind of weird is that Ed Pegg's sequence mentioned by Hans 
Havermann deals with fifth powers, while David Wilson's sequence I 
mention here deals with powers of 5.... Oooo, Spoooky...)


I might as well use this opportunity to point out that I have noticed 
that many times when two OEIS sequences agree for a significant number of 
terms, then first disagree in their mth terms, then the mth terms of both 
sequences are often relatively close, even if the sequences' definitions 
are far apart.
Can this phenomenon be quantified and explained somehow? (Maybe sequences 
often rise exponentially, perhaps?)

thanks,
Leroy Quet