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 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.