1) Let pn denote the number of ways to build a pipe n units long, using segments that are either plastic or metal, and (for each material) come in lengths of 1 unit or 2 units. For example, p1 = 2 since we can use a 1-unit segment that is either plastic or metal, and P2 = 6 since we can use either type of 2-unit segment, or any of the 22 possible ordered choices of 2 segments each having a length of 1 unit. Define po = 1. Determine a recurrence relation for pn. Give a combinatorial proof that your recurrence relation does solve this counting problem. Use your recurrence relation and the method of generating functions to find a formula for pn. Hint: Your final answer should be for every n ≥ 0.] 1 Pn= (1 + √√3)+¹_ (1-√√3)2+1 2√3 1 2√3
1) Let pn denote the number of ways to build a pipe n units long, using segments that are either plastic or metal, and (for each material) come in lengths of 1 unit or 2 units. For example, p1 = 2 since we can use a 1-unit segment that is either plastic or metal, and P2 = 6 since we can use either type of 2-unit segment, or any of the 22 possible ordered choices of 2 segments each having a length of 1 unit. Define po = 1. Determine a recurrence relation for pn. Give a combinatorial proof that your recurrence relation does solve this counting problem. Use your recurrence relation and the method of generating functions to find a formula for pn. Hint: Your final answer should be for every n ≥ 0.] 1 Pn= (1 + √√3)+¹_ (1-√√3)2+1 2√3 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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![1) Let Pn denote the number of ways to build a pipe n units long, using segments that
are either plastic or metal, and (for each material) come in lengths of 1 unit or 2 units.
For example, p1 = 2 since we can use a 1-unit segment that is either plastic or metal,
and p2= 6 since we can use either type of 2-unit segment, or any of the 22 possible
ordered choices of 2 segments each having a length of 1 unit. Define po = 1. Determine
a recurrence relation for pn. Give a combinatorial proof that your recurrence relation
does solve this counting problem. Use your recurrence relation and the method of
generating functions to find a formula for pn.
[Hint: Your final answer should be
Pn=
1
2√3 (1+√3)+¹-(1-√3)+¹
for every n 20.]
2) Let s, denote the number of lists of any length that have the fixed sum of n, and whose
entries come from {1,2,3). For example, s2 = 2 because (1,1) and (2) are the only
such lists; and 84 = 7 because the lists are (3, 1), (1, 3), (2, 2), (2, 1, 1), (1, 2, 1), (1, 1, 2),
and (1, 1, 1, 1). Define so = 1. Determine 81, 83, and s5 by finding all possible lists.
Give a combinatorial proof that sn = $n-1 + $n-2 + $n-3 for every n ≥ 3. Use this
recurrence relation to show that the generating function S(r) for (sn) is](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2Fcc6c6c0f-af76-4aab-8ba5-c65b9cbc2044%2F71whtvs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1) Let Pn denote the number of ways to build a pipe n units long, using segments that
are either plastic or metal, and (for each material) come in lengths of 1 unit or 2 units.
For example, p1 = 2 since we can use a 1-unit segment that is either plastic or metal,
and p2= 6 since we can use either type of 2-unit segment, or any of the 22 possible
ordered choices of 2 segments each having a length of 1 unit. Define po = 1. Determine
a recurrence relation for pn. Give a combinatorial proof that your recurrence relation
does solve this counting problem. Use your recurrence relation and the method of
generating functions to find a formula for pn.
[Hint: Your final answer should be
Pn=
1
2√3 (1+√3)+¹-(1-√3)+¹
for every n 20.]
2) Let s, denote the number of lists of any length that have the fixed sum of n, and whose
entries come from {1,2,3). For example, s2 = 2 because (1,1) and (2) are the only
such lists; and 84 = 7 because the lists are (3, 1), (1, 3), (2, 2), (2, 1, 1), (1, 2, 1), (1, 1, 2),
and (1, 1, 1, 1). Define so = 1. Determine 81, 83, and s5 by finding all possible lists.
Give a combinatorial proof that sn = $n-1 + $n-2 + $n-3 for every n ≥ 3. Use this
recurrence relation to show that the generating function S(r) for (sn) is
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