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Reducing Quadratic Forms by Kneading Sequences
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Barry R. Smith

Department of Mathematical Sciences

Lebanon Valley College

Annville, PA 17003

USA

**Abstract:**

We introduce an invertible operation on finite sequences of positive
integers and call it "kneading". Kneading preserves three invariants of
sequences the parity of the length, the sum of the entries, and one we
call the "alternant". We provide a bijection between the set of sequences
with alternant *a* and parity *s*
and the set of Zagier-reduced indefinite
binary quadratic forms with
discriminant *a*^{2} + (-1)^{s} · 4,
and show that
kneading corresponds to Zagier reduction of the corresponding forms. It
follows that the sum of a sequence is a class invariant of the
corresponding form. We conclude with some observations and conjectures
concerning this new invariant.

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(Concerned with sequences
A000048.)

Received September 17 2014; revised versions received September 30
2014; October 10 2014; November 22 2014. Published in *Journal of
Integer Sequences*, December 13 2014.

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