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=== Pythagorean triples === The equality {{nowrap|1=''c''<sup>2</sup> = ''a''<sup>2</sup> + (''a'' + 1)<sup>2</sup> = 2''a''<sup>2</sup> + 2''a'' + 1}} occurs exactly when {{nowrap|1=2''c''<sup>2</sup> = 4''a''<sup>2</sup> + 4''a'' + 2}} which becomes {{nowrap|1=2''P''<sup>2</sup> = ''H''<sup>2</sup> + 1}} with the substitutions {{nowrap|1=''H'' = 2''a'' + 1}} and {{nowrap|1=''P'' = ''c''}}. Hence the ''n''-th solution is {{nowrap|1=''a<sub>n</sub>'' = {{sfrac|''H''<sub>2''n''+1</sub> β 1|2}}}} and {{nowrap|1=''c<sub>n</sub>'' = ''P''<sub>2''n''+1</sub>}}. The table above shows that, in one order or the other, ''a<sub>n</sub>'' and {{nowrap|1=''b<sub>n</sub>'' = ''a<sub>n</sub>'' + 1}} are {{nowrap|1=''H<sub>n</sub>H''<sub>''n''+1</sub>}} and {{nowrap|1=2''P<sub>n</sub>P''<sub>''n''+1</sub>}} while {{nowrap|1=''c<sub>n</sub>'' = ''H''<sub>''n''+1</sub>''P<sub>n</sub>'' + ''P''<sub>''n''+1</sub>''H<sub>n</sub>''}}.
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