A145508 a(n+1)=a(n)^2+2*a(n)-2 and a(1)=8.

8, 78, 6238, 38925118, 1515164889164158, 2295724641355838227053650177278, 5270351628928392053240255925779522360603268430188068127843838

Offset: 1

Author

Artur Jasinski, Oct 11 2008

Keywords

• nonn
• easy

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))

Crossrefs

A145502-A145510.

Programs

Mathematica

aa = {}; k = 8; Do[AppendTo[aa, k]; k = k^2 + 2 k - 2, {n, 1, 10}]; aa
or
k =7; Table[Floor[((k + Sqrt[k^2 + 4])/2)^(2^(n - 1))], {n, 2, 7}] (*Artur Jasinski*)
NestList[#^2+2#-2&,8,8] (* Harvey P. Dale, Sep 20 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*(9 + sqrt(77)). a(n) = 1 (mod 7).

Recurrence: a(n) = 10*{product {k = 1..n-1} a(k)} - 2 with a(1) = 8.

Product {n = 1..inf} (1 + 1/a(n)) = 10/sqrt(77).

Product {n = 1..inf} (1 + 2/(a(n) + 1)) = sqrt(11/7).

(End)