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Suppose we have the sequence

$$ \left(0, {3\over5}, {4\over5}, {15\over17}, {12\over13},{35\over37},\ldots\right).$$

Is it possible to find an explicit formula $a_n$? I cant seem to find one.

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    $\begingroup$ These pairs are a part of Pythagorean triplets. $\endgroup$ – The Demonix _ Hermit Dec 3 '19 at 16:21
  • $\begingroup$ The denominator is always the hypotenuse and the sequence looks to be strictly increasing. $\endgroup$ – Andrew Chin Dec 3 '19 at 16:27
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    $\begingroup$ @TheDemonix_Hermit Well-spotted! So they appear to be the sines of the acute angles of right triangles with rational sides, in increasing order. $\endgroup$ – saulspatz Dec 3 '19 at 16:27
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If you know the rule to generate your sequence, you may prove by induction that:

$$a_n=\frac{n^2-1}{n^2+1}, n\ge 1$$

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  • $\begingroup$ does not work..... $\endgroup$ – James Dec 3 '19 at 16:54
  • $\begingroup$ It works for your set... $\endgroup$ – Pspl Dec 3 '19 at 16:55
  • $\begingroup$ check a_3 ..... $\endgroup$ – James Dec 3 '19 at 17:02
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    $\begingroup$ $a_3=8/10=4/5$. Similarly, $a_5=24/26=12/13$ etc $\endgroup$ – almagest Dec 3 '19 at 17:07
  • $\begingroup$ @almagest, thanks :) $\endgroup$ – Pspl Dec 3 '19 at 17:24

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