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A243808 Position within the triangular array A226314(n)/A054531(n) of rationals x/y such that x < y, gcd(x,y)=1 and x+y odd.

%I A243808 #29 Nov 09 2024 07:09:30
%S A243808 2,5,7,9,12,14,16,20,23,25,27,29,31,33,35,38,40,44,46,48,52,54,57,59,
%T A243808 61,63,65,67,71,73,77,80,82,84,86,88,90,92,94,96,100,102,104,107,109,
%U A243808 113,119,121,123,125,127,129,131,133,135,138,140
%N A243808 Position within the triangular array A226314(n)/A054531(n) of rationals x/y such that x < y, gcd(x,y)=1 and x+y odd.
%C A243808 If {x,y} are used as the generators of primitive Pythagorean triples (a,b,c) where a=y^2-x^2, b=2x*y and c=x^2+y^2, then the above sequence enumerates all PPT's as a 1-to-1 mapping into the integers.
%H A243808 Frank M Jackson, <a href="/A243808/b243808.txt">Table of n, a(n) for n = 1..57</a>
%H A243808 Lance Fortnow, <a href="http://blog.computationalcomplexity.org/2004/03/counting-rationals-quickly.html">Counting the Rationals Quickly</a>, Computational Complexity Weblog, Monday, March 01, 2004.
%H A243808 Yoram Sagher, <a href="http://www.jstor.org/stable/2324846">Counting the rationals</a>, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.
%e A243808 .  j       {A226314(n),A054531(n)}, 1<=i<=j<=12 and n=i+j(j-1)/2
%e A243808 .  --   ------------------------------------------------------------
%e A243808 .   1:  1,1
%e A243808 .   2:  1,2 2,1
%e A243808 .   3:  1,3 2,3 3,1
%e A243808 .   4:  1,4 3,2 3,4 4,1
%e A243808 .   5:  1,5 2,5 3,5 4,5 5,1
%e A243808 .   6:  1,6 4,3 5,2 5,3 5,6 6,1
%e A243808 .   7:  1,7 2,7 3,7 4,7 5,7 6,7 7,1
%e A243808 .   8:  1,8 5,4 3,8 7,2 5,8 7,4 7,8 8,1
%e A243808 .   9:  1,9 2,9 7,3 4,9 5,9 8,3 7,9 8,9 9,1
%e A243808 .  10:  1,10 6,5 3,10 7,5 9,2 8,5 7,10 9,5 9,10 10,1
%e A243808 .  11:  1,11 2,11 3,11 4,11 5,11 6,11 7,11 8,11 9,11 10,11 11,1
%e A243808 .  12:  1,12 7,6 9,4 10,3 5,12 11,2 7,12 11,3 11,4 11,6 11,12 12,1 .
%e A243808 a(4)=9, as the 4th PPT is generated from the 9th term of the triangular array at index (3,4). This gives (x,y) as (3,4) and it generates the PPT (7,24,25). Conversely the PPT (7,24,25) gives (x,y) = (sqrt((25-7)/2), sqrt((25+7)/2)) = (3,4). It is the 9th term of the triangular array and the 4th term of the enumerating sequence.
%t A243808 ratmap[p_, q_] := (q(q-1)/2+p); mm=20; lst={}; Do[If[OddQ[m+n]&&GCD[m, n]==1, AppendTo[lst, n/m]], {m, 1, mm}, {n, 1, m}]; Sort@Table[ratmap[Numerator[lst[[k]]], Denominator[lst[[k]]]], {k, 1, Length[lst]}]
%Y A243808 Cf. A054531, A169581, A226314.
%K A243808 nonn
%O A243808 1,1
%A A243808 _Frank M Jackson_, Jun 13 2014