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Letting \(P_{n}\) be the \(n\)th Pell number:


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Today I've been playing around with maths looking for patterns and formulae for even powers of Pell numbers.

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Last sequences I published in the OEIS are and I made a GIF that illustrates sequence A282389 so well. โ˜บ

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that \(\sqrt{2}\) is irrational:
Suppose not, i.e. \( \sqrt{2} = \frac{a}{b} \), \( a,b \in \mathbb{Z} \) and \(a,b\) are coprime.
Then \( \left(\frac{a}{b}\right)^2 = 2\). Then \(a^2 = 2b^2\), so \(a^2\) is even. Let \(a = 2c\), so we have \(4c^2 = 2b^2 \implies b^2 = 2c^2\). So \(b\) is even.
Then both \(a\) and \(b\) are divisible by 2, contradicting the earlier proposition.
So \(\sqrt{2}\) can't be expressed as the ratio of two integers, which means it's irrational.


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