A365310 - OEIS

A365310

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

4, 12, 144, 33024, 2147549184, 9223372041149743104, 170141183460469231750134047789593657344, 57896044618658097711785492504343953926975274699741220483192166611388333031424

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

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).

LINKS

Table of n, a(n) for n=0..7.

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).

MATHEMATICA

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

PROG

(Python)

for n in range(0, 8):

print(2**(2**n)+2**(2**(n+1)-1))

CROSSREFS

Cf. A000215, A058891.

Sequence in context: A052598 A230691 A032071 * A222393 A367883 A359655

Adjacent sequences: A365307 A365308 A365309 * A365311 A365312 A365313

KEYWORD

nonn,easy

AUTHOR

César Aguilera, Aug 31 2023

STATUS

approved