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
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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## 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.
Transcribed Image Text:## 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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