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New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile
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Kenneth Edwards and Michael A. Allen

Physics Department

Faculty of Science

Mahidol University

Rama 6 Road

Bangkok 10400

Thailand

**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 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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