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

0,2

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.

References

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

Links

Table of n, a(n) for n=0..14.

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

Formula

a(n) = A377011(n,1)*A377011(n,2)/2.

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

Example

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.

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

Crossrefs

Cf. A377011, A377016, A002315, A078522, A008844.

Sequence in context: A265586 A278725 A186245 * A224429 A289325 A202923

Adjacent sequences: A377014 A377015 A377016 * A377018 A377019 A377020

Keyword

nonn

Author

Miguel-Ángel Pérez García-Ortega, Oct 13 2024

Status

approved