Publication: New combinatorial interpretations of the fibonacci numbers squared, golden rectangle numbers, and jacobsthal numbers using two types of tile
Issued Date
2021-01-01
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ISSN
15307638
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2-s2.0-85103845325
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Mahidol University
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SCOPUS
Bibliographic Citation
Journal of Integer Sequences. Vol.24, No.3 (2021)
Suggested Citation
Kenneth Edwards, Michael A. Allen New combinatorial interpretations of the fibonacci numbers squared, golden rectangle numbers, and jacobsthal numbers using two types of tile. Journal of Integer Sequences. Vol.24, No.3 (2021). Retrieved from: https://repository.li.mahidol.ac.th/handle/123456789/77387
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Title
New combinatorial interpretations of the fibonacci numbers squared, golden rectangle numbers, and jacobsthal numbers using two types of tile
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Abstract
We consider the tiling of an n-board (a board of size n × 1) with squares of unit width and (1, 1)-fence tiles. A (1, 1)-fence tile is composed of two unit-width square sub-tiles separated by a gap of unit width. We show that the number of ways to tile an n-board using unit-width squares and (1, 1)-fence tiles is equal to a Fibonacci number squared when n is even and a golden rectangle number (the product of two consec-utive Fibonacci numbers) when n is odd. We also show that the number of tilings of boards using n such square and fence tiles is a Jacobsthal number. Using combinatorial techniques we prove new identities involving golden rectangle and Jacobsthal numbers. Two of the identities involve entries in two Pascal-like triangles. One is a known triangle (with alternating ones and zeros along one side) whose (n, k)th entry is the number of tilings using n tiles of which k are fence tiles. There is a simple relation between this triangle and the other which is the analogous triangle for tilings of an n-board. These triangles are related to Riordan arrays and we give a general procedure for finding which Riordan array(s) a triangle is related to. The resulting combinatorial interpretation of the Riordan arrays allows one to derive properties of them via combinatorial proof.