A145502 - OEIS

A145502

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

2, 6, 46, 2206, 4870846, 23725150497406, 562882766124611619513723646, 316837008400094222150776738483768236006420971486980606

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OFFSET

1,1

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))).

From Peter Bala, Nov 12 2012: (Start)

The present sequence corresponds to the case x = 3 of the following general remarks. Sequences A145503 through A145510 correspond to the cases x = 4 through x = 11 respectively.

Let x > 2 and let alpha := {x + sqrt(x^2 - 4)}/2. Define a sequence a(n) (which depends on x) by setting a(n) = alpha^(2^(n-1)) + (1/alpha)^(2^(n-1)) - 1. Then it is easy to verify that the sequence a(n) satisfies the recurrence equation a(n+1) = a(n)^2 + 2*a(n) - 2 with the initial condition a(1) = x - 1.

A second recurrence is a(n) = (a(1) + 2)*{Product_{k = 1..n-1} a(k)} - 2.

The following algebraic identity is valid for x > 2:

(x + 1)/sqrt(x^2 - 4) = (1 + 1/(x - 1))*(y + 1)/sqrt(y^2 - 4), where y - 1 = (x - 1)^2 + 2*(x - 1) - 2. Iterating the identity yields the product expansion (x + 1)/sqrt(x^2 - 4) = Product {n = 1..oo} (1 + 1/a(n)).

A second expansion is Product {n = 1..oo} (1 + 2/(a(n) + 1) = sqrt((x + 2)/(x - 2)). For an alternative approach to these identities see the Bala link.

(End)

LINKS

Table of n, a(n) for n=1..8.

Peter Bala, Notes on A145502-A145510.

Daniel Duverney, Irrationality of Fast Converging Series of Rational Numbers, Journal of Mathematical Sciences-University of Tokyo, Vol. 8, No. 2 (2001), pp. 275-316.

Daniel Duverney and Takeshi Kurosawa, Transcendence of infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2002), Article 68.

FORMULA

From Peter Bala, Nov 12 2012: (Start)

a(n) = phi^(2^n) + (1/phi)^(2^n) - 1, where phi := (1 + sqrt(5))/2 is the golden ratio.

a(n) = A001566(n-1) - 1.

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

Product_{n >= 1} (1 + 1/a(n)) = 4/sqrt(5).

Product_{n >= 1} (1 + 2/(a(n) + 1)) = sqrt(5). (End)

From Amiram Eldar, Sep 10 2022: (Start)

a(n) = A000324(n) - 3.

Sum_{n>=1} (-2)^n/a(n) = -1/2 (Duverney, 2001). (End)

Product_{n>=1} (1 + 3/a(n)) = 4 (Duverney and Kurosawa, 2022). - Amiram Eldar, Jan 07 2023