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A372219 Four-column table read by rows: row n is the unique primitive Pythagorean quadruple (a,b,c,d) such that a < (a + b + c - d)/2 = 2n(n + 1) and b = c.

%I A372219 #16 May 19 2024 11:53:21
%S A372219 1,12,12,17,7,30,30,43,17,56,56,81,31,90,90,131,49,132,132,193,71,182,
%T A372219 182,267,97,240,240,353,127,306,306,451,161,380,380,561,199,462,462,
%U A372219 683,241,552,552,817,287,650,650,963,337,756,756,1121,391,870,870,1291,449,992,992,1473
%N A372219 Four-column table read by rows: row n is the unique primitive Pythagorean quadruple (a,b,c,d) such that a < (a + b + c - d)/2 = 2n(n + 1) and b = c.
%C A372219 A Pythagorean quadruple is a quadruple (a,b,c,d) of positive integers such that a^2 + b^2 + c^2 = d^2 with a <= b <= c. Its inradius is (a+b+c-d)/2, which is a positive integer.
%D A372219 Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
%H A372219 Miguel-Ángel Pérez García-Ortega, <a href="/A372219/a372219.pdf">Teorema 10.12</a>
%F A372219 Row n = (a, b, c, d) = (2n^2 - 1, 4n^2 + 6n + 2, 4n^2 + 6n + 2, 6n^2 + 8n + 3).
%e A372219 Table begins:
%e A372219   n=1:    1,  12,    12,    17;
%e A372219   n=2:    7,  30,    30,    43;
%e A372219   n=3:   17,  56,    56,    81;
%e A372219   n=4:   31,  90,    90,   131;
%e A372219   n=5:   49, 132,   132,   193;
%t A372219 cuaternas={};Do[cuaternas=Join[cuaternas,{2n^2-1,4n^2+6n+2,4n^2+6n+2,6n^2+8n+3}],{n,1,35}];cuaternas
%Y A372219 Cf. A372220, A056220 (first column), A002939 (second column), A126587 (fourth column).
%K A372219 nonn,easy,tabf
%O A372219 2,2
%A A372219 _Miguel-Ángel Pérez García-Ortega_, Apr 22 2024