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
Faculty of Science
Rama 6 Road
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
Full version: pdf,
(Concerned with sequences
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.
Journal of Integer Sequences home page