## Approximations of Square Root of 2

A001333 :   Numerators of continued fraction convergents to   $\sqrt {2}$
7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243, 275807, 665857, …

A000129 :   Pell numbers
5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, …

$7/5 \; \approx \; \sqrt {2}$

$7^2 \; \approx \; 5^2 \; \times \; 2$

$7^2 \; = \; 49$
$5^2 \; \times \; 2 \; = \; 50$

$(7 + 5 + 5)/(7 + 5) \; = \; 17/12 \approx \sqrt {2}$

$17^2 \; = \; 289$
$12^2 \; \times \; 2 \; = \; 288$

$(17 + 12 + 12)/(17 + 12) \; = \; 41/29 \approx \sqrt {2}$

$41^2 \; = \; 1681$
$29^2 \; \times \; 2 \; = \; 1682$

$(41 + 29 + 29)/(41 + 29) \; = \; 99/70 \approx \sqrt {2}$

$99^2 \; = \; 9801$
$70^2 \; \times \; 2 \; = \; 9800$

$(99 + 70 + 70)/(99 + 70) \; = \; 239/169 \approx \sqrt {2}$

$239^2 \; = \; 57121$
$169^2 \; \times \; 2 \; = \; 57122$

$(239 + 169 + 169)/(239 + 169) \; = \; 577/408 \approx \sqrt {2}$

$577^2 \; = \; 332929$
$408^2 \; \times \; 2 \; = \; 332928$

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