Journal of Integer Sequences, Vol. 12 (2009), Article 09.1.5

The Pfaffian Transform

Tracale Austin
3824 Teton Pass
Ellenwood, GA 30294

Hans Bantilan
Department of Physics
Princeton University
Princeton, NJ 08544

Eric S. Egge
Department of Mathematics
Carleton College
Northfield, MN 55057

Isao Jonas
Challenge Online Games, Inc.
816 Congress Ave, Suite 1470
Austin, TX 78701

Paul Kory
Department of Mathematics
Indiana University
831 East 3rd Street
Bloomington, IN 47405


We introduce a function on sequences, which we call the Pfaffian transform, using the Pfaffian of a skew-symmetric matrix. We establish several basic properties of the Pfaffian transform, and we use the transfer matrix method to show that the set of sequences with rational generating functions is closed under the Pfaffian transform. We conclude by computing the Pfaffian transform of a variety of sequences, including geometric sequences, the sequence of Fibonacci numbers, the sequence of Pell numbers, the sequence of Jacobsthal numbers, and the sequence of Tribonacci numbers. Throughout we describe a generalization of our results to Pfaffians of skew-symmetric matrices whose entries satisfy a Pascal-like relation.

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(Concerned with sequences A000045 A000073 A000079 A000108 A000244 A001006 and A001045.)

Received June 8 2008; revised version received December 16 2008. Published in Journal of Integer Sequences, December 17 2008.

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