Simplification of Two Papers of Ferri, Faccio, and

D’Amico

Helmut Prodinger Department for Algebra and Discrete Mathematics TU Wien Wiedner Hauptstrasse 8–10/118 A-1040 Wien Austria email: Helmut.Prodinger@tuwien.ac.at

In [1], the authors come up with a “new” numerical triangle, defined recursively by

a_{n,k} = a_{n−1,k} + ∑_{j=0}^{n−1} a_{j,k−1}, n > k, a_{n,n} = a_{n,0} = 1 .  (1)

Complicated expressions are derived for these numbers. However, they are not so “new” at all, since

a_{n,k} = \binom{n + k}{n − k} .  (2)

The second paper [2] is very similar; only the starting values are different: a_{n,0} = n + 1.

This time we find

a_{n,k} = \binom{n + k + 1}{n − k} .  (3)

The proofs (for instance by induction) are completely standard and therefore omitted.

References

[1] G. Ferri, M. Faccio, and A. D’Amico. A new numerical triangle showing links with Fibonacci numbers. The Fibonacci Quarterly, 29:316–321, 1991.

[2] G. Ferri, M. Faccio, and A. D’Amico. Fibonacci numbers and ladder network impedance. The Fibonacci Quarterly, 30:62–67, 1992.

1991 Mathematics Subject Classification. 05A10.

Date: April 27, 1996.