A219164 Recurrence equation a(n+1) = a(n)^4 - 4*a(n)^2 + 2 with a(0) = 5.

5, 527, 77132286527, 35395236908668169265765137996816180039862527

Offset: 0

Author

Peter Bala, Nov 13 2012

Keywords

• nonn
• easy

Comments

Bisection of A003487.

The next term -- a(4) -- has 175 digits. - Harvey P. Dale, Jun 09 2017

Crossrefs

Cf. A001999, A003487, A003501, A219162, A219163, A219165.

Programs

Mathematica

NestList[#^4-4#^2+2&,5,5] (* Harvey P. Dale, Jun 09 2017 *)

Formula

Let alpha = 1/2*(5 + sqrt(21)). Then a(n) = (alpha)^(4^n) + (1/alpha)^(4^n).

a(n) = A003487(2*n) = A003501(4^n).

Product_{n>=0} ((1 + 2/a(n))/(1 - 2/a(n)^2)) = sqrt(7/3).

From Peter Bala, Dec 06 2022: (Start)

a(n) = 2*T(4^n,5/2), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind.

Let b(n) = a(n) - 5. The sequence {b(n)} appears to be a strong divisibility sequence, that is, gcd(b(n),b(m)) = b(gcd(n,m)) for n, m >= 1. (End)