New Seq: Differences between squares are cubes

[zak seidov]

Dear seqfans,

My Qs, plz:

Is a(42)= 7821697850 the last term
(if corresponding q exists it is > ~ 10^10).
As i know (actually i don't) 
there is no theory about a^2-b^2=c^3, right?
Anyway, more terms?

thanks, zak

PS i hope to get some responses before submitting
these two seq's:

%S A1
1,3,6,10,15,17,561,564,423564,1268439,2535189,4223814,6334314,8866689,11820939,15197064,18995064,23214939,27856689,32920314,38405814,44313189,50642439,54153730,57782670,61529259,65393497,69375384,73474920,77692105,82026939,86479422,91049554,95737335,100542765,105465844,110506572,115664949,120940975,126334650,126385350,126385350,7821697850
%N A1
a(1)=1, then a(n) is the smallest integer > a(n-1)
such that a(n)^2-a(n-1)^2 is cube q^3.

%C A1 
Values of q's are (future A2):
2,3,4,5,4,68,15,5640,11265,16890,22515,28140,33765,39390,45015,50640,56265,
61890,67515,73140,78765,84390,71659,74060,76461,78862,81263,83664,86065,88466,
90867,93268,95669,98070,100471,102872,105273,107674,110075,23400,3940000

Note that sequence of q's is not monotonic.
The sequence presumably terminates with
a(42)=7821697850(?)

%e A1 
3^2-1^2=2^3,
6^2-3^2=3^3,
10^2-6^2=4^3,
15^2-10^2=5^3,
17^2-15^2=4^3,
561^2-17^2=68^3.

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zs> From seqfan-owner at ext.jussieu.fr  Thu Apr 10 09:56:42 2008
zs> From: zak seidov <zakseidov at yahoo.com>
zs> Subject: New Seq: Differences between squares are cubes
zs> To: seqfan at ext.jussieu.fr
zs> ....
zs> Is a(42)= 7821697850 the last term
zs> (if corresponding q exists it is > ~ 10^10).
zs> As i know (actually i don't) 
zs> there is no theory about a^2-b^2=c^3, right?
zs> Anyway, more terms?
zs> 
zs> thanks, zak
zs> 
zs> PS i hope to get some responses before submitting
zs> these two seq's:
zs> 
zs> %S A1
zs> 1,3,6,10,15,17,561,564,423564,1268439,2535189,4223814,6334314,8866689,11820939,15197064,18995064,23214939,27856689,32920314,38405814,44313189,50642439,54153730,57782670,61529259,65393497,69375384,73474920,77692105,82026939,86479422,91049554,95737335,100542765,105465844,110506572,115664949,120940975,126334650,126385350,126385350,7821697850
zs> %N A1
zs> a(1)=1, then a(n) is the smallest integer > a(n-1)
zs> such that a(n)^2-a(n-1)^2 is cube q^3.
zs>...

This list is confirmed by a PARI implementation (which didn't find anything beyond 7821697850 so far):

cbrtint(n)={
}

A1(aprev)={
}

{
}