A377253 - OEIS

%I #10 Oct 28 2024 04:10:42

%S 1,3,161,78651,344321281,13556330774163,4803014772895786721,

%T 15315151467227056585562187,439511508577232466840070997207041,

%U 113517180330950376886852596699083807208099,263873290425594499887783413858790664311869763220001,5520415640853364978944256380321540917933730551194875050712027

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

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

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

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

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

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

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

%F a(n) ~ 2^(n-1) * 3^(n^2). - _Vaclav Kotesovec_, Oct 28 2024

%e Illustration of the initial terms: a(0) = 1,

%e a(1) = ( (3 + sqrt(8)) + (3 - sqrt(8)) )/2 = 3,

%e a(2) = ( (9 + sqrt(80))^2 + (9 - sqrt(80))^2 )/2 = 161,

%e a(3) = ( (27 + sqrt(728))^3 + (27 - sqrt(728))^3 )/2 = 78651,

%e a(4) = ( (81 + sqrt(6560))^4 + (81 - sqrt(6560))^4 )/2 = 344321281,

%e a(5) = ( (243 + sqrt(59048))^5 + (243 - sqrt(59048))^5 )/2 = 13556330774163,

%e ...

%o (PARI) {a(n) = round( cosh( n*acosh(3^n) ) )}

%o (PARI) {a(n) = round( ((3^n + sqrt(9^n-1))^n + (3^n - sqrt(9^n-1))^n)/2 )}

%o (PARI) {a(n) = sum(k=0, n\2, binomial(n, 2*k) * (3^n)^(n-2*k) * (9^n-1)^k)}

%o (PARI) {a(n) = 3^(n^2) * sum(k=0, n\2, binomial(n, 2*k) * (1 - 1/9^n)^k)}

%o (PARI) {a(n) = polcoeff( (1 - 3^n*x)/(1 - 2*3^n*x + x^2 +x*O(x^n)), n)}

%o for(n=0,12, print1(a(n),", "))

%Y Cf. A197320.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 27 2024