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**
A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four
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Michael D. Hirschhorn

School of Mathematics and Statistics

UNSW

Sydney 2052

Australia

James A. Sellers

Department of Mathematics

Penn State University

University Park, PA 16802

USA

**Abstract:**

In 2001, Andrews and Lewis utilized an identity of F. H. Jackson to
derive some new partition generating functions as well as identities
involving some of the corresponding partition functions. In particular,
for 0 < *a* < *b* < *k*, they defined
*W*_{1}(*a*, *b*; *k*; *n*) to be the number of
partitions of *n* in which the parts are congruent to *a* or
*b* mod *k* and such that, for any *j*, *kj* +
*a* and *kj* + *b* are not both parts. Our primary goal
in this note is to prove that *W*_{1}(1,3;4;27*n* + 17) ≡
0 (mod 3) for all *n* ≥ 0. We prove this result using
elementary generating function manipulations and classic results from
the theory of partitions.

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(Concerned with sequences
A000122
A000203
A001935
A010054
A070048
A121455.)

Received June 29 2014; revised versions received September 2 2014; September 4 2014.
Published in *Journal of Integer Sequences*, September 4 2014.

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