A377011 - OEIS

%I #44 Nov 04 2024 16:29:29

%S 1,0,1,7,24,25,41,840,841,239,28560,28561,1393,970224,970225,8119,

%T 32959080,32959081,47321,1119638520,1119638521,275807,38034750624,

%U 38034750625,1607521,1292061882720,1292061882721,9369319,43892069261880,43892069261881,54608393,1491038293021224,1491038293021225

%N Table read by rows: row n is the unique primitive Pythagorean triple whose short leg is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

%C A002315(n) is a NSW number (named after Newman, Shanks, and Williams) and is an integer h that solves the Diophantine equation 2*n^2 = h^2 + 1.

%C Row n=0 is included by convention and corresponds to the Pythagorean triple 1^2 + 0^2 = 1^2.

%C The hypotenuses of all these Pythagorean triples are perfect squares.

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

%F row(n) = (A002315(n), A078522(n+1), A008844(n)).

%F row(n) = (2*A114336(3n-2) + 1, (A114336(3n))^2 - 1, (A114336(3n))^2) for all n > 0.

%e Table begins:

%e   n=0:      1,         0,         1;

%e   n=1:      7,        24,        25;

%e   n=2:     41,       840,       841;

%e   n=3:    239,     28560,     28561;

%e   n=4:   1393,    970224,    970225;

%e   n=5:   8119,  32959080,  32959081;

%t t[n_]:=t[n]= Module[{a, b},a=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;b=(a^2-1)/2;{a,b,b+1}];ternas={};Do[ternas=Join[ternas,FullSimplify[t[n]]],{n,0,10}];ternas

%Y Cf. A002315 (short leg), A078522 (long leg), A008844 (hypotenuse), A001652 (inradius), A377016 (semiperimeter), A377017 (area), A001653 (square root of the hypotenuse), A114336.

%K nonn,easy,tabf

%O 0,4

%A _Miguel-Ángel Pérez García-Ortega_, Oct 12 2024