A110407 -id:A110407 - cp's OEIS frontend

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

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Oct 11 2008

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]

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

Cf. A145502-A145510. A003010, A002812.

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 *)

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)

A178497 5 followed by A145503.

Original entry on oeis.org

5, 3, 13, 193, 37633, 1416317953, 2005956546822746113, 4023861667741036022825635656102100993, 16191462721115671781777559070120513664958590125499158514329308740975788033

Offset: 0

Roger L. Bagula, May 28 2010

nonn

The first 6 entries of the sequence are primes.

The Fermat numbers F(.)=A000215(.) obey equations with flipped signs of the +-2 if compared with the a(n) (see the formula section): F(n) = Prod{i=0,...,n-1} F(i)+2 and F(n) = F(n-1) *(F(n-1)-2)+2. - Vladimir Shevelev, Dec 08 2010

Cf. A000215, A110407, A178428, A178427.

Clear[a, n];
a[ -1] := 1; a[0] := 5;
a[n_] := a[n] = Product[a[i], {i, 0, n - 1}] - 2;
Table[a[n], {n, 0, 10}]

a(n)= ( product_{i=0..n-1} a(i)) -2, n>=1.

a(n) = a(n-1)*( a(n-1)+2 )-2, n>=2. - Vladimir Shevelev, Dec 08 2010

Definition simplified by the Assoc. Eds. of the OEIS, Jun 07 2010

cp's OEIS Frontend

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

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