A365310 a(n) = 2^(2^n) + 2^(2^(n+1)-1).

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Offset: 0

Author

César Aguilera, Aug 31 2023

Keywords

• nonn
• easy

Comments

a(n) is the long leg of the Pythagorean triangle whose short leg is the n-th Fermat number, A000215(n), and whose hypotenuse is a(n) + 1.

[A000215(n), a(n), a(n) + 1] is a primitive Pythagorean triple of the form [2*k + 1, 2*k^2 + 2*k, 2*k^2 + 2*k + 1] where k = A058891(n).

Crossrefs

Cf. A000215, A058891.

Programs

Mathematica

Table[2^(2^n) + 2^(2^(n + 1) - 1), {n, 0, 7}] (* Paul F. Marrero Romero, Jan 13 2024 *)

Python

for n in range(0,8):
    print(2**(2**n)+2**(2**(n+1)-1))

Formula

a(n) = A000215(n) + A058891(n+1) - 1.

a(n) = sqrt(Integral_{x=1..A000215(n)} (x^3-x) dx).

a(n) = 2*A058891(n)^2 + 2*A058891(n).

sqrt(a(n) + (a(n)+1)) = sqrt((a(n)+1)^2 - a(n)^2) = A000215(n).