#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

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

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

2. Find f(5), f(6)

f(5) = 10, f(6) = 134

3. What is F(10^6)? How long did it take?
What is f(10^6)? How long did it take?

Too large to compute, takes forever
F(10^5) has over 29,900 digits

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?

f(n + 1000) - f(n + 999) - 5 * f(n)

ope(N) = N^1000 - N^999 - 5

5. Characterize the sequences that satisfy a homogeneous recurrence of order 0

f(n) = 0

6. Can you prove d(n)/n! -> 1/e

d(n) = n! - Sum(binomial(n,k)*d(n-k), k=1..n)
d(n) = n! - Sum((-1)^(k-1)*(n!/k!), k=1..n)
d(n) = Sum((-1)^k*(n!/k!))
d(n) ~ n!/e
d(n)/n! --> 1/e

7. What is the OEIS number of this sequence?

The OEIS number of the sequence 1,1,2,4,10,26,76,232,764,2620,... is A000085.

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

w(n) = w(n - 1) + (n - 1) * w(n - 2)
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)

9. Look up the syntax of ZeilbergerRecurrence

ZeilbergerRecurrence(T, n, k, f, l..u)