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This is related to How many sequences of rational squares are there, all of whose differences are also rational squares?

Are there infinite sequences $a_n$ of rational cubes whose first differences are positive squares?

Fixing $a_n$ leads to finding points on the elliptic curve $y^2=x^3-a_n^3$. If it is of positive rank there will be infinitely many choices for $a_{n+1}$.

Is there an explicit construction that avoids finding points on curves?

What about infinite sequence of integer cubes -- in this case fixing $a_n$ leads to finding integral points on elliptic curves and these are finite.

A possible generalization might be:

Are there infinite sequences of numbers of type $X$ (e.g. triangular, Fibonacci) whose first differences are positive squares?

I am mainly interested in explicit constructions like the case for $X=\square$.

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Careful with the generalization - there is an infinite sequence of numbers of the form $n(n+1)(2n+1)/6$ whose first differences are positive squares. –  Gerry Myerson Dec 2 '11 at 10:12
    
First differences of Fibonacci numbers are Fibonacci numbers, and J.H.E. Cohn has proved earlier on that the only square Fibonacci numbers are 1 and 144. Bugeaud-Mignotte-Siksek generalized this recently by proving that the only perfect powers in the Fibonacci sequence are 1, 8 and 144. –  François Brunault Dec 5 '11 at 12:36
    
Sorry, I misread your last question, so my comment doesn't seem really helpful. –  François Brunault Dec 5 '11 at 12:37
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François, they need not be consecutive, e.g. $F_{13}-F_{11}=2^{4} \cdot 3^{2}$ –  joro Dec 5 '11 at 12:54
    
For Fibonacci numbers, see mathoverflow.net/questions/84797 –  Max Alekseyev Dec 8 '13 at 19:02
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