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[Appeared in J. Difference Equations and Applications
14 (2008) , 1119 - 1126 (Special issue in honor of
Gerry Ladas' 70th Birthday)]

Written: Dec. 5, 2007.

Dedicated to Gerry Ladas, on his 70th Birthday

We have a marvelous proof of Gessel's lattice-walk conjecture, but
the margins (we mean our computers) are not wide (we mean large) enough to contain
it. Let's hope that we won't need to wait 350 years for another "proof".

Note added June 30, 2008: We only had to wait 6 months!
See the
exciting new paper, coauthored by Christoph Koutschan and the
authors of the present paper.

## Mathematica Packages

This article is accompanied by the following Mathematica
packages (written by Manuel Kauers):

## Maple Packages

This article is accompanied by the following Maple
packages (written by Doron Zeilberger):

## Sample Input and Output

- To get a completely computer-generated (and rigorous (of course!)) proof
of Kreweras' famous theorem, using the quasi-holonomic ansatz of our paper

the input
produces the
output

(Note: Kreweras' problem does belong to the (fully) holonomic ansatz,
but the (pure) recurrence, in n, is much larger.
See the
pure recurrence equation (in n) .)
- To get a completely computer-generated (and rigorous (of course!)) proof
of Marni Mishna's Second Class of Walks,

the input
produces the
output
- To get a completely computer-generated (and rigorous (of course!)) proof
of Marni Mishna's Fourth Class of Walks,

the input
produces the
output
- To get a completely computer-generated (and rigorous (of course!)) proof
of Marni Mishna's Fifth Class of Walks,

the input
produces the
output
- To get a completely computer-generated (and rigorous (of course!)) proof
of Marni Mishna's Sixth Class of Walks,

the input
produces the
output
- To get a completely computer-generated (and rigorous (of course!)) proof
of Marni Mishna's Seventh Class of Walks, the
input
produces the
output
- To get a completely computer-generated (and rigorous (of course!)) proof
of Marni Mishna's Eighth Class of Walks,

the input
produces the
output
- To get a completely computer-generated (and rigorous (of course!)) proof
of Marni Mishna's Ninth Class of Walks,

the input
produces the
output

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