Periodicity of Euler Numbers in Modular Arithmetic

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The Euler numbers are integers that arise in the series expansion of the hyperbolic secant function
around the origin:
. The plot above indicates that the sequence
is periodic in
for any integer
. Incidentally, the sequence
is periodic with respect to
.
Contributed by: Oleksandr Pavlyk (March 2011)
Open content licensed under CC BY-NC-SA
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As of August 27, 2009, Stan Wagon informed the author, quoting Herbert Wilf, that the following proof is well known and published in the book by S. K. Lando, Lectures on Generating Functions, Providence, RI: AMS, 2003.
The formal generating function of Euler numbers has the simple continued fraction
;
likewise, the formal generating function for the absolute value of the Euler numbers is
.
Hence, the formal generating functions of the sequences and
have a terminating continued fraction expansion, and thus are rational functions in
. It is well known that a rational generating function gives rise to a periodic sequence in modular arithmetic.
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