*Forma,* Vol. 19 (No. 4), pp. 355-366, 2004

*Original Paper*

## Generalized Binet Formulas, Lucas Polynomials, and Cyclic Constants

Jay Kappraff^{1}* and Gary W. Adamson^{2}

^{1}Department of Mathematics, New Jersey Institute of Technology, Newark, NJ 07102, U.S.A.

^{2}P.O. Box 124571, San Diego, CA 92112-4571, U.S.A.

*E-mail address: kappraff@verizon.net

(Received March 11, 2005; Accepted March 15, 2005)

**Keywords: **
Binet's Theorem, Silver Means, Fibonacci Sequence, Pell Sequence, Gauss' Theorem, Pell-Lucas Sequence, Fibonacci Polynomials, Lucas Polynomials

**Abstract. **
Generalizations of Binet's theorem are used to produce generalized Pell sequences from two families of silver means. These Pell sequences are also generated from the family of Fibonacci polynomials. A family of Pell-Lucas sequences are also generated from the family of Lucas polynomials and from another generalization of Binet's formula. A periodic set of cyclic constants are generated from the Lucas polynomials. These cyclic constants are related to the Gauss-Wantzel proof of the constructibility by compass and straightedge of regular polygons.

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