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A362218 Three-column array read by rows: row n gives the unique ordered primitive Pythagorean triple (a,b,c) with a such that (b+c)/a = n.

%I A362218 #37 Apr 30 2023 18:11:34
%S A362218 3,4,5,8,15,17,5,12,13,12,35,37,7,24,25,16,63,65,9,40,41,20,99,101,11,
%T A362218 60,61,24,143,145,13,84,85,28,195,197,15,112,113,32,255,257,17,144,
%U A362218 145,36,323,325,19,180,181,40,399,401
%N A362218 Three-column array read by rows: row n gives the unique ordered primitive Pythagorean triple (a,b,c) with a<b such that (b+c)/a  = n.
%C A362218 Given an ordered primitive Pythagorean triple (a,b,c) with a<b, we call (b+c)/a the median radial proportion. This number gives us the proportion (r_b)/r, where r_b is the exinradius associated to the greater cathetus of the Pythagorean triangle (a,b,c) and r is the inradius of the same triangle.
%C A362218 For n>=3 there exists a unique ordered primitive Pythagorean triple such that (b+c)/a = n.
%C A362218 For n odd, the triple is {n, (n^2-1)/2, (n^2+1)/2}.
%C A362218 For n even, the triple is { 2*n, n^2-1, n^2+1 }.
%D A362218 J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.
%H A362218 Miguel-Ángel Pérez García-Ortega, <a href="/A362218/a362218.pdf">Radial proportions</a> (in Spanish).
%F A362218 T(n,1) = A022998(n).
%F A362218 T(n,2) = A066830(n).
%F A362218 T(n,3) = A228564(n).
%F A362218 a(6*k-3) = 2*k+1;
%F A362218 a(6*k-2) = ((2*k+1)^2 - 1)/2;
%F A362218 a(6*k-1) = ((2*k+1)^2 + 1)/2;
%F A362218 a(6*k) = 4*(k+1);
%F A362218 a(6*k+1) = 4*(k+1)^2 - 1;
%F A362218 a(6*k+2) = 4*(k+1)^2 + 1.
%e A362218 Irregular array begins:
%e A362218   n=3:   3,  4,  5;
%e A362218   n=4:   8, 15, 17;
%e A362218   n=5:   5, 12, 13;
%e A362218   n=6:  12, 35, 37;
%e A362218   n=7:   7, 24, 25;
%e A362218   ...
%e A362218 Row n=3 is (3,4,5) and has (b+c)/a = (4+5)/3 = 3.
%e A362218 Row n=4 is (8,15,17) and has (b+c)/a = (15+17)/8 = 4.
%t A362218 k=50;
%t A362218 ternas={{n," ",a,b,c," ",r," "," γ2 "," ",s," ",rb}};Do[If[Mod[t,2]==0,ternas=Join[ternas,{{t," ",2t,t^2-1,t^2+1," ",t-1," ",t," ",t(t+1)," ",t(t-1)}}],ternas=Join[ternas,{{t," ",t,(t^2-1)/2,(t^2+1)/2," ",(t-1)/2," ",t," ",(t(t+1))/2," ",(t(t-1))/2}}]],{t,3,k+2}]
%t A362218 MatrixForm[Transpose[ternas]]
%Y A362218 Cf. A022998 (short leg), A066830 (long leg), A228564 (hypotenuse).
%K A362218 nonn,tabf
%O A362218 3,1
%A A362218 _Miguel-Ángel Pérez García-Ortega_, Apr 11 2023
%E A362218 Edited by _N. J. A. Sloane_, Apr 30 2023