A219163 Recurrence equation a(n+1) = a(n)^4 - 4*a(n)^2 + 2 with a(0) = 4.

4, 194, 1416317954, 4023861667741036022825635656102100994

Offset: 0

Author

Peter Bala, Nov 13 2012

Keywords

• nonn
• easy

Comments

Bisection of A003010.

a(4) has 147 digits and a(5) has 586 digits. - Harvey P. Dale, Mar 03 2020

Crossrefs

Cf. A001999, A003010, A003500, A219162, A219164, A219165.

Programs

Mathematica

NestList[#^4-4#^2+2&,4,5] (* Harvey P. Dale, Mar 03 2020 *)

PARI

a(n)={if(n==0,4,a(n-1)^4-4*a(n-1)^2+2)} \\ Edward Jiang, Sep 11 2014

Formula

Let alpha = 2 + sqrt(3). Then a(n) = (alpha)^(4^n) + (1/alpha)^(4^n).

a(n) = A003010(2*n) = A003500(4^n).

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

From Peter Bala, Dec 06 2022: (Start)

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

Let b(n) = a(n) - 4. 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)