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Pascal-like triangles and Fibonacci-like sequences

Published online by Cambridge University Press:  23 January 2015

H. Matsui ,
D. Minematsu ,
T. Yamauchi  and
R. Miy Adera
H. Matsui
Affiliation:
Kwansei Galruin High School, Nishinomiya City, Japan, e-mail:miyadera172000@yahoo.co.jp
D. Minematsu
Affiliation:
Kwansei Galruin High School, Nishinomiya City, Japan, e-mail:miyadera172000@yahoo.co.jp
T. Yamauchi
Affiliation:
Kwansei Galruin High School, Nishinomiya City, Japan, e-mail:miyadera172000@yahoo.co.jp
R. Miy Adera
Affiliation:
Kwansei Galruin High School, Nishinomiya City, Japan, e-mail:miyadera172000@yahoo.co.jp
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Extract

In [1] and [2] we demonstrated how Pascal-like triangles arose from the probabilities associated with the various outcomes of a particular game (see Definition 1 below). It was also shown that they could be considered as generalisations of Pascal's triangle. In this article we show how Fibonacci-like sequences arise from our Pascal-like triangles, and demonstrate the existence of simple relationships between these Fibonacci-like sequences and the Fibonacci sequence itself. In addition we will investigate a generalisation of the binomial coefficients that appears when considering an extended version of the game. We start by describing this game.

Type
Articles
Information
The Mathematical Gazette , Volume 94 , Issue 529 , March 2010 , pp. 27 - 41
Copyright
Copyright © The Mathematical Association 2010

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References

1. Miyadera, R., Hashiba, S., Hashiba, T., Nakagawa, Y., Matsui, H., Yamanchi, T., Sakaguchi, M. and Minematsu, D., Pascal-like triangles and Sierpinski-like gaskets, Visual mathematics art and science electric journal of ISIS-Symmetry, 9(1) (2007).Google Scholar
2. Miyadera, R., Hashiba, S. and Minematsu, D., Mathematical theory of magic fruits – interesting patterns of fractions, Archimedes' Laboratory: http://www-archimedes-lab.org/fraction_patt/Magic_fruits.html Google Scholar
3. The On-line Encyclopedia ofInteger Sequences: http://www.research.att.com/~njas/sequences/ Google Scholar
4. Matsui, H., Saita, N., Kawata, K., Sakuraya, Y. and Miyadera, R., Elementary Problems, B-1019, Fibonacci Quarterly, 44.3 (2006).Google Scholar