Has Anyone Seen This Formula?

[Alex Violette]

Hello SeqFans,

I wanted to reach out to see if anyone recognizes this formula I self discovered in high school(before 2020) for Pythagorean Triples where T_n=(n^2+n)/2:

(sqrt(8*T_n+1))^2+(4*T_n)^2=(4*T_n+1)^2

Has anyone seen this or anything like it? I to date have not found this anywhere really or at least not in this form. I've attached a screenshot of it from a snippet of a little paper I wrote up in high school.

Best,

Alex Violette

[Giovanni Resta]

The formula in itsel trivially works with any x since

sqrt(8x+1)^2 is just 8x+1.

and clearly 8x+1 + 16x^2 = 16x^2 + 8x + 1 = (4x+1)^2.

When x is a triangular number, then 8x+1 is a square so this correspond to
a Pythagorean triple.

The triples you get in this way are exactly those that can be written in a simpler way as 

{k, (k^2-1)/2, (k^2+1)/2} for an odd k > 1.

This  was known for a long time, indeed it is the so called "Fibonacci's method". 
See the first method on https://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples

Giovanni

[brad klee]

Another Harm.On.ica artifact has been produced: 

https://github.com/bradleyklee/Artifacts/tree/main/04-nicomachus-theorem

The Nicomachus theorem is pretty spectacular and nowadays there are 

cool videos of it on youtube. That's going to draw more attention, even 

though Pythagorean triples are also great to know. 

--Brad 

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