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A328972
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Denominator of the fraction (hypotenuse - difference of legs) / (sum of legs - hypotenuse) of the n-th primitive Pythagorean triangle.
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2
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1, 2, 3, 3, 3, 5, 4, 5, 5, 4, 7, 5, 7, 6, 5, 4, 9, 7, 7, 9, 7, 11, 8, 7, 6, 5, 11, 9, 9, 8, 7, 13, 6, 11, 9, 10, 13, 8, 11, 15, 13, 11, 10, 9, 11, 8, 15, 7, 13, 12, 11, 11, 17, 9, 13, 8, 17, 13, 11, 15, 11, 10, 13, 19, 17, 14, 8, 13, 12, 11, 19, 13, 17, 10, 9, 15, 14, 21, 13
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OFFSET
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1,2
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COMMENTS
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A328971(n) / a(n) should contain all reduced fractions between 1 and sqrt(2) + 1 without duplicates.
A328971(n) (numerators) is built from the difference between the length of the hypotenuse (A020882) and the difference between the two legs (A120682) of the n-th primitive Pythagorean triangle.
a(n) is built from the difference between the sum of the length of the legs (A120681) and the hypotenuse of the n-th primitive Pythagorean triangle.
Then both numbers are divided by their GCD to get the reduced fraction.
All primitive Pythagorean triangles are sorted first on hypotenuse, then on long leg.
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LINKS
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EXAMPLE
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For n=13 we need the 13th primitive Pythagorean triangle:
36,77,85
^ ^ We calculate the difference between the two small numbers: 77-36=41.
^ To get our numerator we subtract 41 from the hypotenuse length: 85-41=44.
^ ^ Then we calculate the sum of the two small numbers: 36+77=113.
^ We subtract 85 from this sum to get the denominator: 113-85=28.
This gives us the fraction 44/28 and in reduced form 11/7.
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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