A145503 a(n+1) = a(n)^2+2*a(n)-2 and a(1)=3.

3, 13, 193, 37633, 1416317953, 2005956546822746113, 4023861667741036022825635656102100993

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)).

Essentially the same as A110407. [R. J. Mathar, Mar 18 2009]

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..11

Crossrefs

Cf. A145502-A145510. A003010, A002812.

Programs

Mathematica

aa = {}; k = 3; Do[AppendTo[aa, k]; k = k^2 + 2 k - 2, {n, 1, 10}]; aa
(* or *)
k = 2; Table[Floor[((k + Sqrt[k^2 + 4])/2)^(2^(n - 1))], {n, 2, 7}]
NestList[#^2+2#-2&,3,10] (* Harvey P. Dale, Feb 01 2018 *)

Formula

From Peter Bala, Nov 12 2012: (Start)

a(n) = alpha^(2^(n-1)) + (1/alpha)^(2^(n-1)) - 1, where alpha := 2 + sqrt(3).

a(n) = A003010(n-1) - 1. a(n) = 2*A002812(n-1) - 1.

Recurrence: a(n) = 5*(Product {k = 1..n-1} a(k)) - 2 with a(1) = 3.

Product_{n >= 1} (1 + 1/a(n)) = 5/6*sqrt(3).

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

(End)