## Tiling a 2 × n Checkerboard with L-Shaped Pieces### Problem OverviewYou have a large $2 \times n$ checkerboard and you aim to cover its $2n$ squares using the available pieces. Let $a_n$ represent the number of legal tilings of a $2 \times n$ board. The pieces used are gray L-shaped tiles that cover 3 squares. Below is an example of a legal tiling:(Insert diagram: A rectangular board with L-shaped tiles covering specific areas.)### Tasks#### (a) Initial Values- Determine the initial values $a_1$, $a_2$, $a_3$.- Note that not all board sizes can be fully covered, so some values may be zero.#### (b) Recurrence Relation- Find a recurrence relation satisfied by the sequence $a_n$.#### (c) Generating Function- Identify the rational function that serves as a generating function for these numbers.#### (d) Exact Formula- Derive an exact formula for $a_{3m}$ for $m = 1, 2, 3, \ldots$.### Diagram ExplanationThe given diagram showcases a $2 \times 3$ board covered by two L-shaped tiles. Each L-shaped tile covers precisely 3 squares, illustrating a possible configuration where the board is completely tiled without any overlaps or gaps. When devising solutions, consider the orientation and placement of these L-shaped tiles as you explore the possible arrangements for various board lengths.

Given that : an= 2×n In-order to find the values of a1,a2 and a3; a1=2*1=2 a2=2*2=4 a3=2*3=6 In-order to find the recurrence relation, an= a1+a2+a3=a3+a2+a12+4+6=6+4+212=12The Rational Functions are, 2:4:6 which in terms of, 1:2:3 i.e.,. a1:a2:a3 In-order to find the exact formula of a3m=2×3m m=1,2,3,....a3×1=2×3=6a3×2=2×6=12a3×3=2×9=18.....

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You have a large 2 x n checkerboard, and you want to cover its 2n squares using the available pieces. Let an board.The available pieces are gray L-shaped pieces that cover 3 squares For example, this is a legal tiling the number of legal tilings of a 2 × n (a) Determine the initial values a1, a2, a3 (Not all board sizes covered, so some values could be zero.) an be (b) Find a recurrence relation that is satisfied by the sequence an. (c) Find the rational function that is a generating function for these numbers. (d) Find an exact formula for a3m for m = 1,2, 3, . ..

You have a large 2 x n checkerboard, and you want to cover its 2n squares using the available pieces. Let an board.The available pieces are gray L-shaped pieces that cover 3 squares For example, this is a legal tiling the number of legal tilings of a 2 × n (a) Determine the initial values a1, a2, a3 (Not all board sizes covered, so some values could be zero.) an be (b) Find a recurrence relation that is satisfied by the sequence an. (c) Find the rational function that is a generating function for these numbers. (d) Find an exact formula for a3m for m = 1,2, 3, . ..

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

## Tiling a 2 × n Checkerboard with L-Shaped Pieces

### Problem Overview
You have a large $2 \times n$ checkerboard and you aim to cover its $2n$ squares using the available pieces. Let $a_n$ represent the number of legal tilings of a $2 \times n$ board. The pieces used are gray L-shaped tiles that cover 3 squares. Below is an example of a legal tiling:

(Insert diagram: A rectangular board with L-shaped tiles covering specific areas.)

### Tasks

#### (a) Initial Values
- Determine the initial values $a_1$, $a_2$, $a_3$.
- Note that not all board sizes can be fully covered, so some values may be zero.

#### (b) Recurrence Relation
- Find a recurrence relation satisfied by the sequence $a_n$.

#### (c) Generating Function
- Identify the rational function that serves as a generating function for these numbers.

#### (d) Exact Formula
- Derive an exact formula for $a_{3m}$ for $m = 1, 2, 3, \ldots$.

### Diagram Explanation
The given diagram showcases a $2 \times 3$ board covered by two L-shaped tiles. Each L-shaped tile covers precisely 3 squares, illustrating a possible configuration where the board is completely tiled without any overlaps or gaps.

When devising solutions, consider the orientation and placement of these L-shaped tiles as you explore the possible arrangements for various board lengths.

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