#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:

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

Q1. Convert the following recurrence to explicit form
    6*f(n) + 12*f(n + 1) + 18*f(n + 3) = 0

A1. 
    f(n + 3) = -12 * f(n + 1) - 6*f(n) 

Q2. Find f(5) and f(6) for f(n) = 3*f(n - 1) - 4*f(n-2)
	f(0) = 1, f(1) = -2
A2. 
	f(5) = 10 
	f(6) = 134

Q3. 
    What is F(10^6), how long did it take? 
    What is f(10^6), how long did it take? 


A3. 
	for f(10^6) maple complains too many levels of recursion
	for F(10^6) maple took too long and i had to abort

Q4. 
	Consider the sequence that is defined by the recurrence
	f(n + 1000) = f(n + 999) + 5 * f(n)
	What is the opeartor ope(N) such that
	ope(N) f(n) = 0? 

A4. 
	f(n + 1000) - f(n + 999) - 5 * f(n) = 0
	ope(N) = N^1000 - N^999 - 5


Q5. 
	Characterize the sequences that satisfy a (homog.) recurrence of order zero. [16:00]
	f(n+1) = c * f(n)

A5. 
	f(n + 1) = c * f(n)
	The sequence should be x^k

Q6. can u prove d(n)/n! -> 1/e
A6. 

Q7. What is the OEIS A-number for the sequence [40:22]
A7. A85
	
Q8. Find the operators in N and N^(-1) annihilated by w(n)
A8. 
	w(n + 2) - w(n + 1) - (n + 1)*w(n) = 0

	OPE(n, N) = N^2 - N - (n + 1)
	multiply N^(-1) we have
	N^(-1) * (N^2 - N - (n + 1))
	 = N - 1 - n * N^(-1)

Q9. Look up the syntax of ZeilbergerRecurrence 
A9. 
	https://www.maplesoft.com/support/help/Maple/view.aspx?path=SumTools%2FHypergeometric%2FZeilberger