derivation of recurrence for Sylvester’s sequence

Let us begin with the product:

$$a_n=1+\prod _{i=0}^{n-1}{a}_i$$

Adding $1$ to $n$ and manipulating the result:

$a_{n+1}$ $=$ $1+{\displaystyle \prod _{i=0}^n}{a}_i$

$=$ $1+a_n{\displaystyle \prod _{i=0}^{n-1}}{a}_i$

$=$ $1+a_n(a_n-1)=1+{(a_n)}^2-a_n$

Title	derivation of recurrence for Sylvester’s sequence
Canonical name	DerivationOfRecurrenceForSylvestersSequence
Date of creation	2013-03-22 15:48:27
Last modified on	2013-03-22 15:48:27
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	4
Author	rspuzio (6075)
Entry type	Derivation
Classification	msc 11A55