#ATTENDANCE QUIZ FOR LECTURE 16 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p16
#with an attachment called
#p16FirstLast.txt
#(e.g. p16DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Oct. 30, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 9

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER


# Question 1. Convert the following recurrence to explicit form: 

# 6*f(n) + 12*f(n+1) + 18*f(n+3) = 0

# ANSWER: f(n+3) = -1/2 * f(n+1) - 1/3 * f(n)  

# Question 2. Find f(5) and f(6) of f(n)=3*f(n-1)-4*f(n-2), f(0)=1, f(1)=-2

# ANSWER: 
# f(5)=3*(-26)-4*(-22)=-78+88=10

# f(6)=3*10-4*(-26)=30+104=134

# Question 3. What is F(10^6)? How long did it take? --> It took over 4 minutes, didn't get a value.
#  What is f(10^6)? How long did it take? --> Too many levels of recursion. Still not getting a value.

# Question 4. Consider the sequence that is defined by the recurrence
# f(n+1000) = f(n+999) + 5*f(n) 
# What is the operator ope(N) such that ope(n) f(n) = 0

# ANSWER: 

# f(n+1000) = f(n+999) + 5*f(n)
# f(n+1000) - f(n+999) - 5*f(n) = 0
# N^1000 - N^999 - 5N

# Question 5. Characterize the sequence(s) that satisfy a homogeneous recurrence of order zero. 

# ANSWER: 
# f(n) = c where c is some constant

# Question 6. Can you prove that d(n)/n! -> 1/e3

# ANSWER: 
# d(n)=n*d(n-1)+(-1)^n
# d(n)=n*((n-1) * d(n-2) + (-1)^(n-1)) + (-1)^n
# ...
# d(n)=n! * Sum((-1)^i/i!, i = 0..n)

# The Taylor expansion of e^x = Sum(x^i/i!, i = 0..infinity).

# Then, when we plug in -1 for x, we get that
# the limit as n goes to infinity for d(n) / n! = Sum((-1)^i/i!, i = 0..infty) = e^-1 = 1/e.

# Question 7. What is the OEIS A number of this sequence?

# ANSWER: 
# A85

# Question 8. Find the operators in N and N^-1 annihilated by w(n):

# ANSWER: 
# w(n) - w(n - 1) - (n - 1) * w(n - 2) = 0
# w(n+2) - w(n+1) - (n + 1) * w(n) = 0
# ope = N^2 - N - (n + 1)

# Question 9. Do ?SumTools[Hypergeometric][ZeilbergerRecurrence]

# ANSWER: 
# Parameters:
# T-hypergeometric term of n and k
# n-name
# k-name
# En-name; denote the shift operator with respect to n
# f-name of the recurrence function
# l..u-range for k
# 'Zpair'-list of two elements specifying a Z-pair for T