Journal of Integer Sequences, Vol. 17 (2014), Article 14.11.8

Reducing Quadratic Forms by Kneading Sequences

Barry R. Smith
Department of Mathematical Sciences
Lebanon Valley College
Annville, PA 17003


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 a2 + (-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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