A Generalization Of The Fibonacci Formulae

1982 ◽  
Vol 75 (8) ◽  
pp. 664-667
Author(s):  
Lucille A. Kelly
Keyword(s):  
Liberal Arts ◽  
Fibonacci Numbers ◽  
Recursive Sequences ◽  
Triangular Numbers ◽  
Mathematics Class ◽  
Liberal Arts Mathematics

Students are surprised by the magic within the Fibonacci numbers, and they find that with some experimentation they can discover some of this magic. The following generalization arose from a discussion in a liberal arts mathematics class that had been learning about patterns in sets of numbers such as the counting numbers, even numbers, odd numbers, and the square and triangular numbers and finally forming recursive sequences by creating recursive formulae.

scholarly journals Natural Mathematics, the Fibonacci Numbers and Aesthetics in Art

2019 ◽  
Vol 17 ◽  
pp. 248-254
Author(s):  
Anthony G Shannon ◽  
Seamus A. Power
Keyword(s):  
Liberal Arts ◽  
Fibonacci Numbers ◽  
Visual Aesthetics

The Mathematics of beauty and beauty in mathematics are important ingredients in learning in the liberal arts. The Fibonacci numbers play an important and useful role in this. This paper seeks to present and illustrate a grounding of visual aesthetics in natural mathematical principles, centered upon the Fibonacci numbers. The specific natural mathematical principles investigated are the Fibonacci numbers, the Fibonacci Spiral, and the Cosmic Bud.

Grammarmathically Speaking

1966 ◽  
Vol 59 (7) ◽  
pp. 640-645
Author(s):  
King W. Jamison
Keyword(s):  
Liberal Arts ◽  
Long Time ◽  
Liberal Arts Mathematics

Much has been said which identifies mathematics as an art. Few students, teachers of mathematics, and laymen would deny the esthetic value of mathematics, and tradition labels mathematics as an art. As one example, the curriculum of the medieval university included mathematics among the seven liberal arts. Mathematics has also been thought of as a science for a long time. Now, aided by modern communications, even the most uninterested person associates mathematics with science and scientists. It is therefore comparatively easy to accept the notion that mathematics is an art and that mathematics is a science.

scholarly journals Cassini- and Catalan-like formulas for polygonal numbers and simplex numbers

SURG Journal ◽  
2012 ◽  
Vol 5 (2) ◽  
pp. 37-43
Author(s):  
Thomas Jeffery
Keyword(s):  
Fibonacci Numbers ◽  
Triangular Numbers ◽  
Polygonal Numbers

Cassini’s formula and Catalan’s formula are two results from the theory of Fibonacci numbers. This article derives results similar to these, however instead of applying to Fibonacci numbers, they are applied to polygonal numbers and simplex numbers. Triangular numbers are considered first. We then generalize to polygonal and simplex numbers. For polygonal numbers the properties of determinants are used to simplify calculations. For simplex numbers Pascal’s Theorem is used.

scholarly journals ARBITRARY ORDER RECURSIVE SEQUENCES AND ASSOCIATED CONTINUED FRACTIONS

2017 ◽  
Vol 13 (2) ◽  
pp. 7147-7154
Author(s):  
Anthony G Shannon ◽  
Charles K Cook b ◽  
Rebecca A. Hillman c
Keyword(s):  
Continued Fractions ◽  
Arbitrary Order ◽  
Fibonacci Numbers ◽  
Recursive Sequences ◽  
Essential Idea ◽  
Associated Continued Fractions ◽  
Generalized Continued Fractions

The essential idea in this paper it to generalize and synthesize some of the pioneering ideas of Bernstein, Lucas and Horadam on generalizations of basic and primordial Fibonacci numbers and their arbitrary order generalizations and their relation to generalized continued fractions with matrices as the unifying elements.

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