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A328971 Numerator of the fraction (hypotenuse - difference of legs) / (sum of legs - hypotenuse) of the n-th primitive Pythagorean triangle.

%I A328971 #29 Nov 16 2019 04:10:00
%S A328971 2,3,5,4,7,7,5,9,6,7,9,11,11,7,8,9,11,13,8,13,15,13,9,10,11,12,15,10,
%T A328971 17,11,12,15,13,17,19,11,17,13,19,17,19,12,13,14,21,15,19,16,21,13,14,
%U A328971 23,19,16,23,17,21,14,25,23,16,17,25,21,23,15,19,16,17,18,23,27,25,19,20,16,17,23,18
%N A328971 Numerator of the fraction (hypotenuse - difference of legs) / (sum of legs - hypotenuse) of the n-th primitive Pythagorean triangle.
%C A328971 a(n) / A328972(n) should contain all reduced fractions between 1 and sqrt(2) + 1 without duplicates.
%C A328971 a(n) 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.
%C A328971 A328972(n) (denominators) 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.
%C A328971 Then both numbers are divided by their GCD to get the reduced fraction.
%C A328971 All primitive Pythagorean triangles are sorted first on hypotenuse, then on long leg.
%H A328971 S. Brunner, <a href="/A328971/b328971.txt">Table of n, a(n) for n = 1..10000</a>
%H A328971 S. Brunner, <a href="https://pastebin.com/6BbfTGAh">List for n = 0..5000 together with the primitive Pythagorean triangles </a>
%e A328971 For n=2 we need the 2nd primitive Pythagorean triangle:
%e A328971 5,12,13
%e A328971 ^  ^    We calculate the difference between the two small numbers: 12-5=7.
%e A328971       ^ And to get our numerator we subtract 7 from the hypotenuse length: 13-7=6.
%e A328971 ^  ^    Then we calculate the sum of the two small numbers: 5+12=17.
%e A328971       ^ We subtract 13 from this sum to get the denominator: 17-13=4.
%e A328971 This gives us the fraction 6/4, and in reduced form 3/2.
%Y A328971 Denominators: A328972.
%K A328971 frac,nonn
%O A328971 1,1
%A A328971 _S. Brunner_, Nov 01 2019