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A372220 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 A372220 #13 May 19 2024 11:54:07
%S A372220 17,20,20,33,31,42,42,67,49,72,72,113,71,110,110,171,97,156,156,241,
%T A372220 127,210,210,323,161,272,272,417,199,342,342,523,241,420,420,641,287,
%U A372220 506,506,771,337,600,600,913,391,702,702,1067,449,812,812,1233,511,930,930,1411,577,1056,1056,1601
%N A372220 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 A372220 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 A372220 Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
%H A372220 Miguel-Ángel Pérez García-Ortega, <a href="/A372220/a372220.pdf">Teorema 10.13</a>
%F A372220 Row n = (a, b, c, d) = (2n^2 + 4n + 1, 4n^2 + 2n, 4n^2 + 2n, 6n^2 + 4n + 1).
%e A372220 Table begins:
%e A372220   n=2:   17,   20,    20,    33;
%e A372220   n=3:   31,   42,    42,    67;
%e A372220   n=4:   49,   72,    72,   113;
%e A372220   n=5:   71,  110,   110,   171;
%e A372220   n=6:   97,  156,   156,   241;
%t A372220 cuaternas={};Do[cuaternas=Join[cuaternas,{2n^2+4n+1,4n^2+2n,4n^2+2n,6n^2+4n+1}],{n,2,35}];cuaternas
%Y A372220 Cf. A372219, A056220 (first column), A002943 (second column), A080859 (fourth column).
%K A372220 nonn,easy,tabf
%O A372220 2,1
%A A372220 _Miguel-Ángel Pérez García-Ortega_, Apr 22 2024