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\def\pFq#1#2#3#4#5#6{\pFqnoargs{#1}{#2}\biggl(\begin{matrix}%
{#3}\kern.707em{#4}\\{#5}%
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\noindent
Let $a_n$ denote the number of ways in which a chess rook can move from a
corner cell to the opposite corner cell of an $n \times n \times n$
three-dimensional chessboard, assuming that the piece moves closer to the goal
cell at each step. We describe the computer-driven \emph{discovery and
proof\/} of the fact that the generating series $G(x)= \sum_{n \geq 0}
a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function:
\[
G(x) =  1 + 6 \cdot \int_0^x \frac{ \,\pFq21{1/3}{2/3}{2}
  {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.
\]


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