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Motzkin Numbers

A quiz covering the properties, applications, and mathematical relationships of Motzkin numbers, including generating functions, recurrence relations, asymptotic behavior, and connections to other important number sequences.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is the generating function for Motzkin numbers?

  1. $$1 + x + x^2 + x^3 + \cdots$$
  2. $$1 + x + 2x^2 + 4x^3 + \cdots$$
  3. $$1 + x + 3x^2 + 9x^3 + \cdots$$
  4. $$1 + x + 4x^2 + 16x^3 + \cdots$$
Question 2 Multiple Choice (Single Answer)

What is the recurrence relation for Motzkin numbers?

  1. $$M_n = M_{n-1} + M_{n-2}$$
  2. $$M_n = M_{n-1} + 2M_{n-2}$$
  3. $$M_n = M_{n-1} + 3M_{n-2}$$
  4. $$M_n = M_{n-1} + 4M_{n-2}$$
Question 3 Multiple Choice (Single Answer)

What is the value of $$M_5$$?

  1. 5
  2. 10
  3. 15
  4. 20
Question 4 Multiple Choice (Single Answer)

What is the asymptotic formula for Motzkin numbers?

  1. $$M_n \sim \frac{1}{n^{3/2}}\left(\frac{3}{2}\right)^n$$
  2. $$M_n \sim \frac{1}{n^{5/2}}\left(\frac{3}{2}\right)^n$$
  3. $$M_n \sim \frac{1}{n^{7/2}}\left(\frac{3}{2}\right)^n$$
  4. $$M_n \sim \frac{1}{n^{9/2}}\left(\frac{3}{2}\right)^n$$
Question 5 Multiple Choice (Single Answer)

What are Motzkin numbers used for?

  1. Counting lattice paths
  2. Counting triangulations
  3. Counting permutations
  4. Counting graphs
Question 6 Multiple Choice (Single Answer)

What is the connection between Motzkin numbers and Catalan numbers?

  1. $$M_n = 2C_n$$
  2. $$M_n = 3C_n$$
  3. $$M_n = 4C_n$$
  4. $$M_n = 5C_n$$
Question 7 Multiple Choice (Single Answer)

What is the connection between Motzkin numbers and Fibonacci numbers?

  1. $$M_n = F_{2n}$$
  2. $$M_n = F_{2n+1}$$
  3. $$M_n = F_{2n+2}$$
  4. $$M_n = F_{2n+3}$$
Question 8 Multiple Choice (Single Answer)

What is the connection between Motzkin numbers and Stirling numbers of the second kind?

  1. $$M_n = S(n,n)$$
  2. $$M_n = S(n,n+1)$$
  3. $$M_n = S(n+1,n)$$
  4. $$M_n = S(n+1,n+1)$$
Question 9 Multiple Choice (Single Answer)

What is the connection between Motzkin numbers and Delannoy numbers?

  1. $$M_n = D(n,n)$$
  2. $$M_n = D(n,n+1)$$
  3. $$M_n = D(n+1,n)$$
  4. $$M_n = D(n+1,n+1)$$
Question 10 Multiple Choice (Single Answer)

What is the connection between Motzkin numbers and Schröder numbers?

  1. $$M_n = S_n$$
  2. $$M_n = S_{n+1}$$
  3. $$M_n = S_{n+2}$$
  4. $$M_n = S_{n+3}$$
Question 11 Multiple Choice (Single Answer)

What is the connection between Motzkin numbers and Narayana numbers?

  1. $$M_n = N_n$$
  2. $$M_n = N_{n+1}$$
  3. $$M_n = N_{n+2}$$
  4. $$M_n = N_{n+3}$$
Question 12 Multiple Choice (Single Answer)

What is the connection between Motzkin numbers and Catalan-Motzkin numbers?

  1. $$M_n = CM_n$$
  2. $$M_n = CM_{n+1}$$
  3. $$M_n = CM_{n+2}$$
  4. $$M_n = CM_{n+3}$$
Question 13 Multiple Choice (Single Answer)

What is the connection between Motzkin numbers and Touchard numbers?

  1. $$M_n = T_n$$
  2. $$M_n = T_{n+1}$$
  3. $$M_n = T_{n+2}$$
  4. $$M_n = T_{n+3}$$
Question 14 Multiple Choice (Single Answer)

What is the connection between Motzkin numbers and Eulerian numbers?

  1. $$M_n = A_n$$
  2. $$M_n = A_{n+1}$$
  3. $$M_n = A_{n+2}$$
  4. $$M_n = A_{n+3}$$
Question 15 Multiple Choice (Single Answer)

What is the connection between Motzkin numbers and Bell numbers?

  1. $$M_n = B_n$$
  2. $$M_n = B_{n+1}$$
  3. $$M_n = B_{n+2}$$
  4. $$M_n = B_{n+3}$$
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