A377017 Area of the unique primitive Pythagorean triple whose short leg is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

0, 84, 17220, 3412920, 675761016, 133797385260, 26491207202460, 5245125232676784, 1038508304885968560, 205619399242324129860, 40711602541676078766516, 8060691683852625858745320, 1595976241800278270688414120, 315995235184771245126273789084, 62565460590342906257639745449100

Offset: 0

Author

Miguel-Ángel Pérez García-Ortega, Oct 13 2024

Keywords

• nonn

Comments

a(0)=0 is included by convention. This corresponds to the Pythagorean triple 1^2 + 0^2 = 1^2.

All terms in this sequence are divisible by 84.

Examples

			For n=2, the short leg is A002315(2) = 41 and the long leg is A008844(2)-1 = 840 so the area is then a(2) = 41*840/2 = 17220.
		

References

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

Links

• Index entries for linear recurrences with constant coefficients, signature (204,-1190,204,-1).

Crossrefs

Cf. A377016, A002315, A078522, A008844.

Programs

Mathematica

s[n_]:=s[n]=Module[{a, b}, a=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2; b=(a^2-1)/2; {(a*b)/2}]; areas={}; Do[areas=Join[areas, FullSimplify[s[n]]], {n, 0, 17}]; areas

Formula

a(n) = A002315(n)*(A008844(n)-1)/2.

G.f.: 84*x*(1 + x)/((1 - 198*x + x^2)*(1 - 6*x + x^2)). - Andrew Howroyd, Oct 14 2024