A197320 a(n) = cosh(n*arccosh(2^n)).

1, 2, 31, 2024, 522241, 536215712, 2198218022911, 36024948845706368, 2361111184527977349121, 618964706995596541734949376, 649035559893095618486323487178751, 2722257150515888128204116425527951075328, 45671917999814457716384401535256546748378644481

Offset

0,2

Links

Table of n, a(n) for n=0..12.

Formula

a(n) = ( (2^n + sqrt(4^n-1))^n + 1/(2^n + sqrt(4^n-1))^n )/2.

a(n) = 2^(n^2) * ( (1 + sqrt(1-1/4^n))^n + (1 - sqrt(1-1/4^n))^n )/2.

a(n) = 2^(n^2) * Sum_{k=0..floor(n/2)} C(n,2*k) * (1 - 1/4^n)^k.

a(n) = [x^n] (1 - 2^n*x)/(1 - 2*2^n*x + x^2), where [x^n] F(x) denotes the coefficient of x^n in F(x).

a(n) ~ 2^(n^2 + n - 1). - Vaclav Kotesovec, Oct 28 2024

Example

Illustration of the initial terms: a(0) = 1,
a(1) = ( (2 + sqrt(3)) + (2 - sqrt(3)) )/2 = 2,
a(2) = ( (4 + sqrt(15))^2 + (4 - sqrt(15))^2 )/2 = 31,
a(3) = ( (8 + sqrt(63))^3 + (8 - sqrt(63))^3 )/2 = 2024,
a(4) = ( (16 + sqrt(255))^4 + (16 - sqrt(255))^4 )/2 = 522241,
a(5) = ( (32 + sqrt(1023))^5 + (32 - sqrt(1023))^5 )/2 = 536215712, ...

Mathematica

Table[Cosh[n ArcCosh[2^n]], {n, 0, 15}]//Round (* Harvey P. Dale, Aug 14 2019 *)

Prog

(PARI) {a(n)=round(cosh(n*acosh(2^n)))}
(PARI) {a(n)=round(((2^n+sqrt(4^n-1))^n + 1/(2^n+sqrt(4^n-1))^n)/2)}
(PARI) {a(n)=sum(k=0, n\2, binomial(n, 2*k)*(2^n)^(n-2*k)*(4^n-1)^k)}
(PARI) {a(n)=2^(n^2)*sum(k=0, n\2, binomial(n, 2*k)*(1-1/4^n)^k)}
(PARI) {a(n)=polcoeff((1 - 2^n*x)/(1 - 2*2^n*x + x^2 +x*O(x^n)), n)}

Crossrefs

Cf. A377253.

Sequence in context: A358567 A134646 A368354 * A239332 A350940 A244441

Adjacent sequences: A197317 A197318 A197319 * A197321 A197322 A197323

Keyword

nonn

Author

Paul D. Hanna, Oct 28 2011

Status

approved