A145504 a(n+1)=a(n)^2+2*a(n)-2 and a(1)=4.

4, 22, 526, 277726, 77132286526, 5949389624883225721726, 35395236908668169265765137996816180039862526, 1252822795820745419377249396736955608088527968701950139470082687906021780162741058825726

Offset: 1

Author

Artur Jasinski, Oct 11 2008

Keywords

• nonn

Comments

General formula for a(n+1)=a(n)^2+2*a(n)-2 and a(1)=k+1 is a(n)=Floor[((k + Sqrt[k^2 + 4])/2)^(2^((n+1) - 1))

The next term (a(9)) has 175 digits. - Harvey P. Dale, Nov 16 2013

Crossrefs

A145502-A145510. A003487.

Programs

Mathematica

NestList[#^2+2#-2&,4,7] (* Harvey P. Dale, Nov 16 2013 *)

Formula

From Peter Bala, Nov 12 2012: (Start)

a(n) = alpha^(2^(n-1)) + (1/alpha)^(2^(n-1)) - 1, where alpha := 1/2*(5 + sqrt(21)).

a(n) = A003487(n-1) - 1.

Recurrence: a(n) = 6*{product {k = 1..n-1} a(k)} - 2 with a(1) = 4.

Product {n = 1..inf} (1 + 1/a(n)) = 2/7*sqrt(21).

Product {n = 1..inf} (1 + 2/(a(n) + 1)) = sqrt(7/3).

(End)

Extensions

One additional term (a(8)) from Harvey P. Dale, Nov 16 2013