cp's OEIS Frontend

A384329 Table read by rows: row n is the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000217(n) and its long leg and hypotenuse are consecutive natural numbers, n >= 0.

%I A384329 #19 Jul 13 2025 17:25:16
%S A384329 -1,0,1,1,0,1,5,12,13,11,60,61,19,180,181,29,420,421,41,840,841,55,
%T A384329 1512,1513,71,2520,2521,89,3960,3961,109,5940,5941,131,8580,8581,155,
%U A384329 12012,12013,181,16380,16381,209,21840,21841,239,28560,28561,271,36720,36721,305,46512,46513,341,58140,58141
%N A384329 Table read by rows: row n is the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000217(n) and its long leg and hypotenuse are consecutive natural numbers, n >= 0.
%C A384329 Row n = 0 and n = 1 are included by convention and correspond to the Pythagorean triples (-1)^2 + 0^2 = 1^2 and 1^2 + 0^2 = 1^2.
%H A384329 José Miguel Blanco Casado and Miguel-Ángel Pérez García-Ortega, <a href="/A383834/a383834.pdf">El Libro de las Ternas Pitagóricas</a>
%F A384329 row(n) = (2*T(n) - 1, 2*T(n)*(T(n) - 1), 2*T(n)*(T(n) - 1) + 1) where T(n) = A000217(n).
%e A384329   n=0:     -1,     0,     1;
%e A384329   n=1:      1,     0,     1;
%e A384329   n=2:      5,    12,    13;
%e A384329   n=3:     11,    60,    61;
%e A384329   ...
%t A384329 a=Table[(n(n+1))/2,{n,0,18}];Apply[Join,Map[{2#-1,2#^2-2#,2#^2-2#+1}&,a]]
%Y A384329 Cf. A000217, A165900 (short leg), A062392 (semiperimeter), A384498 (sum of the legs).
%K A384329 sign,easy,tabf
%O A384329 0,7
%A A384329 _Miguel-Ángel Pérez García-Ortega_, May 26 2025