# Finding an explicit formula for a sequence

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.

• These pairs are a part of Pythagorean triplets. – The Demonix _ Hermit Dec 3 '19 at 16:21
• The denominator is always the hypotenuse and the sequence looks to be strictly increasing. – Andrew Chin Dec 3 '19 at 16:27
• @TheDemonix_Hermit Well-spotted! So they appear to be the sines of the acute angles of right triangles with rational sides, in increasing order. – saulspatz Dec 3 '19 at 16:27

$$a_n=\frac{n^2-1}{n^2+1}, n\ge 1$$
• $a_3=8/10=4/5$. Similarly, $a_5=24/26=12/13$ etc – almagest Dec 3 '19 at 17:07