# [seqfan] Re: guessing doubling formulas for sequences

A.N.W.Hone A.N.W.Hone at kent.ac.uk
Wed Mar 11 12:00:49 CET 2009

```If a_n is an elliptic divisibility sequence, so a_1=1, a_2,a_3,a_4 are integers with a_2|a_4 and the
recurrence is

a_{n+4}a_n = A a_{n+3}a_{n+1}+Ba_{n+2}^2

where the coefficients are A = a_2^2 and B= - a_1 a_3 = -a_3, then the doubling formula is

a_2 a_{2n}=a_n(a_{n+2}a_{n-1}^2-a_{n-2}a_{n+1}^2.

This is an integer sequence with a_m|a_n whenever m|n.

Andy

________________________________________
From: seqfan-bounces at list.seqfan.eu [seqfan-bounces at list.seqfan.eu] On Behalf Of Robert Israel [israel at math.ubc.ca]
Sent: 10 March 2009 21:58
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: guessing doubling formulas for sequences

On Tue, 10 Mar 2009, Georgi Guninski wrote:

> can someone please give examples of doubling formulas for sequences that
> are not rational (probably exponential)?

For example, a_n = n^n (for n >= 2) has the doubling formula

a_{2n} = a_n^(2+2*ln(2)/LambertW(ln(a_n)))

a_n = n^2 + n + 1 has the doubling formula

a_{2n} = 4 a_n - 2 - sqrt(4 a_n - 3)

Cheers,
Robert Israel

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