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A380821 Length of the shorts leg in the unique primitive Pythagorean triple whose inradius is A000032(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

%I A380821 #20 Jul 14 2025 22:52:17
%S A380821 5,3,7,9,15,23,37,59,95,153,247,399,645,1043,1687,2729,4415,7143,
%T A380821 11557,18699,30255,48953,79207,128159,207365,335523,542887,878409,
%U A380821 1421295,2299703,3720997,6020699,9741695,15762393,25504087,41266479,66770565,108037043
%N A380821 Length of the shorts leg in the unique primitive Pythagorean triple whose inradius is A000032(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
%H A380821 Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, <a href="/A380821/a380821.pdf">El Libro de las Ternas Pitagóricas</a>, Preprint 2025.
%F A380821 a(n) = 2*A000032(n) + 1.
%e A380821  n=0:      5,    12,    13;
%e A380821  n=1:      3,     4,     5;
%e A380821  n=2:      7,    24,    25;
%e A380821  n=3:      9,    40,    41.
%e A380821 This sequence is the first column.
%t A380821 a=Table[LucasL[n],{n,0,15}];Apply[Join,Map[{2#+1,2#^2+2#,2#^2+2#+1}&,a]]
%Y A380821 Cf. A380823 (semiperimeter), A380824 (area), A000032 (inradius), A386201 (long legs).
%K A380821 nonn,easy
%O A380821 0,1
%A A380821 _Miguel-Ángel Pérez García-Ortega_, Feb 04 2025