Journal of Integer Sequences, Vol. 24 (2021), Article 21.3.8

New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile

Kenneth Edwards and Michael A. Allen
Physics Department
Faculty of Science
Mahidol University
Rama 6 Road
Bangkok 10400


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 consecutive 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.

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(Concerned with sequences A000045 A000073 A000124 A000930 A001045 A001654 A002620 A003269 A003600 A006498 A007598 A015518 A017817 A059259 A059260 A071921 A079291 A089928 A114620 A123521 A157897 A158909 A335964.)

Received September 9 2020; revised version received September 12 2020; March 4 2021; March 5 2021. Published in Journal of Integer Sequences, March 10 2021.

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