A377253 a(n) = cosh( n*arccosh(3^n) ).

1, 3, 161, 78651, 344321281, 13556330774163, 4803014772895786721, 15315151467227056585562187, 439511508577232466840070997207041, 113517180330950376886852596699083807208099, 263873290425594499887783413858790664311869763220001, 5520415640853364978944256380321540917933730551194875050712027

Offset

0,2

Links

Table of n, a(n) for n=0..11.

Formula

a(n) = cosh( n*arccosh(3^n) ).

a(n) = ( (3^n + sqrt(9^n-1))^n + (3^n - sqrt(9^n-1))^n )/2.

a(n) = 3^(n^2) * ( (1 + sqrt(1 - 1/9^n))^n + (1 - sqrt(1 - 1/9^n))^n )/2.

a(n) = 3^(n^2) * Sum_{k=0..floor(n/2)} binomial(n,2*k) * (1 - 1/9^n)^k.

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * (3^n)^(n-2*k) * (9^n - 1)^k.

a(n) = [x^n] (1 - 3^n*x)/(1 - 2*3^n*x + x^2).

a(n) ~ 2^(n-1) * 3^(n^2). - Vaclav Kotesovec, Oct 28 2024

Example

Illustration of the initial terms: a(0) = 1,
a(1) = ( (3 + sqrt(8)) + (3 - sqrt(8)) )/2 = 3,
a(2) = ( (9 + sqrt(80))^2 + (9 - sqrt(80))^2 )/2 = 161,
a(3) = ( (27 + sqrt(728))^3 + (27 - sqrt(728))^3 )/2 = 78651,
a(4) = ( (81 + sqrt(6560))^4 + (81 - sqrt(6560))^4 )/2 = 344321281,
a(5) = ( (243 + sqrt(59048))^5 + (243 - sqrt(59048))^5 )/2 = 13556330774163,
...

Prog

(PARI) {a(n) = round( cosh( n*acosh(3^n) ) )}
(PARI) {a(n) = round( ((3^n + sqrt(9^n-1))^n + (3^n - sqrt(9^n-1))^n)/2 )}
(PARI) {a(n) = sum(k=0, n\2, binomial(n, 2*k) * (3^n)^(n-2*k) * (9^n-1)^k)}
(PARI) {a(n) = 3^(n^2) * sum(k=0, n\2, binomial(n, 2*k) * (1 - 1/9^n)^k)}
(PARI) {a(n) = polcoeff( (1 - 3^n*x)/(1 - 2*3^n*x + x^2 +x*O(x^n)), n)}
for(n=0, 12, print1(a(n), ", "))

Crossrefs

Cf. A197320.

Sequence in context: A303099 A302950 A173128 * A157440 A157559 A157586

Adjacent sequences: A377250 A377251 A377252 * A377254 A377255 A377256

Keyword

nonn

Author

Paul D. Hanna, Oct 27 2024

Status

approved