UMass Online
92.419 Intro to Mathematica
Kenneth M. Levasseur
Department of Mathematical Sciences
University of Massachusetts Lowell
Lowell, MA 01854

The Fibonomial Triangle

Subject

Mathematics

Topic

The Fibonomial Triangle - is a triangular array of integers that shares many of the facinating properties of Pascal's Triangle. The basis for the Fibonomial Triangle is the well known Fibonacci Sequence (1,1,2,3,5,8,13,21,34,55.... - each number is the sum of the previous two). Starting with the defining recurrence relation of the sequence,f(n) = f(n-1) + f(n-2), powers of Fibonacci numbers can be expressed in a similar form. For example,

f(n)^2 = 2f(n-1)^2 +2f(n-2)^2 -f(n-3)^2

The coefficients of these equations can be arranged in triagular form:

1
1 1
2 2 -1
3 6 -3 -1
etc.

Reference(s)

1. A. S. DiDomenico, From Fibonacci Numbers to Geometric Sequences and the Binet Formula by Way of the Golden Ratio!,"Mathematics Teacher (May, 1997), 386-389.
2. T. H. Garland, Fascinating Fibonaccis--Mystery and Magic in Numbers,Dale Seymour Publications, 1987.
3. T. M. Green and C. L. Hamberg, Pascal's Triangle, Dale Seymour Publications, Palo Alto, CA 1986.
4. D. K. Hathaway and S. L. Brown, Fibonacci Powers and a Fascinating Triangle, The College Mathematic Journal, (March, 1997), 124-128.
5. Theoni Pappas, The Joy of Mathematics: Discovering Mathematics All Around You, San Carlos, CA: World Wide Pub./Tetra, 1989.

Project Idea(s)

Hathaway & Brown, in [4], introduced the Fibonomial Triangle. Possible projects: derive and implemented the construction of the triangle. Compare it with Pascal's triangle and looked at some generalizations three dimensional versions of the Fibonomial triangles, other similar sequences.

Prerequisite Mathematics

A course in discrete mathematics or linear algebra would be useful.

Required Programming Level

Moderate. The key may be to represent the grid of triangles in an efficient way.

Key Words

Fibonacci, Recursion, Pascal's Triangle, Lucus Numbers

Reviewer

K. M. Levasseur (mailto:kenneth_levasseur@uml.edu)

Archive

None