# 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 |