<s>[INST] Is this sequence following some rule: $(1+3)^2 + (1+2)^2=(2+3)^2$ , $(1+3+5)^2 + (3+4+5)^2=(4+5+6)^2$ , ...? I found its interesting but don't know whether there is any rule that is governing its next term.
The sequence is given as follows:
$$(1+3)^2 + (1+2)^2=(2+3)^2$$
$$(1+3+5)^2 + (3+4+5)^2=(4+5+6)^2$$
$$(1+3+5+7)^2 + (6+7+8+9)^2=(7+8+9+10)^2$$
$$(1+3+5+7+9)^2 + (10+11+12+13+14)^2=(11+12+13+14+15)^2$$
$$\dots$$
$$\text{And so on...}$$
I know how it's coming,
i.e, Pythagorean triplets
For eg.:$(1+3)^2 + (1+2)^2=(2+3)^2$ = ($3,4,5$) Pythagorean triplet
But my question is "Does it following any rule for its next term" ?
 [/INST] A simple induction gives $\left(\frac{n^3+n}{2}\right)^2$ for the right-hand side. On the left-hand side we have $(n^2)^2 + \left(\frac{n^3+n}{2}- n\right)^2$. Since
$$(n^2)^2 + \left(\frac{n^3+n}{2}- n\right)^2 = n^4 + \left(\frac{n^3+n}{2}\right)^2 - 2\frac{n^4+n^2}{2} + n^2 = \left(\frac{n^3+n}{2}\right)^2,$$
they are the same.
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