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Table read by rows: row n is the unique primitive Pythagorean quadruple (a,b,c,d) such that (a + b + c - d)/2 = 2^n - 1 and a = b = 2^n.
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%I #19 Sep 19 2024 09:29:39

%S 4,4,7,9,8,8,31,33,16,16,127,129,32,32,511,513,64,64,2047,2049,128,

%T 128,8191,8193,256,256,32767,32769,512,512,131071,131073,1024,1024,

%U 524287,524289,2048,2048,2097151,2097153,4096,4096,8388607,8388609,8192,8192,33554431,33554433,16384,16384,134217727,134217729

%N Table read by rows: row n is the unique primitive Pythagorean quadruple (a,b,c,d) such that (a + b + c - d)/2 = 2^n - 1 and a = b = 2^n.

%C 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 Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

%H Miguel-Ángel Pérez García-Ortega, <a href="/A371556/a371556.pdf">Contando y calculando cuaternas pitagórcias</a>.

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (-1, -1, -1, 6, 6, 6, 6, -8, -8, -8, -8).

%F Row n = (a, b, c, d) = (2^n, 2^n, 2^(2n - 1) - 1, 2^(2n - 1) + 1).

%e Table begins:

%e n=2: 4, 4, 7, 9;

%e n=3: 8, 8, 31, 33;

%e n=4: 16, 16, 127, 129;

%e n=5: 32, 32, 511, 513;

%e n=6: 64, 64, 2047, 2049;

%t cuaternas={};Do[cuaternas=Join[cuaternas,{2^n,2^n,2^(2n-1)-1,2^(2n-1)+1}],{n,2,35}];cuaternas

%Y Cf. A371559.

%K nonn,easy,tabf

%O 2,1

%A _Miguel-Ángel Pérez García-Ortega_, Mar 27 2024