#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+k)=b1*f(n+k-1)+...+bk*f(n)

f(n+3)=(-12*f(n+1)-6*f(n))/18
f(n+3)=(-2/3)*f(n+1)-(1/3)*f(n)

2) Find f(5) and f(6)

f(5)=3*f(4)-4*f(3)=3*(-26)-4*(-22)=10
f(6)=3*f(5)-4*f(4)=3*10-4*(-26)=134

3) What is F(10^6)? How long did it take?
I don't know, was taking too long
What is f(10^6)? How long did it take?
Fail, too many levels of recursion.

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

The operator is N^1000-N^999+5.

5) Characterize the sequence(s) that satisfy a homog. recurrence of order zero.
They are all not recursive if they are order zero. That means that a(n) does not depend on a(n-1), or a(n-2), or anything.

6) Can you prove that d(n)/n! -> 1/e
The above is equal to d(n)*e/n!=1. d(n) is the number of permutations with no fixed points. n! is all total permutations.
I don't think I know how to do a strong proper proof for this. Empirically though, I can do this:

evalf([seq(d(i)*exp(1)/i!, i = 2..20)]);
  [1.359140914, 0.9060939426, 1.019355686, 0.9967033370, 1.000478728, 0.9999393868, 1.000006804, 0.9999993135, 1.000000063, 0.9999999945, 1.000000000, 0.9999999999, 0.9999999999, 0.9999999999, 0.9999999999, 0.9999999999, 0.9999999999, 0.9999999999, 0.9999999999]

As we can see it does approach 1 very closely, implying that it is indeed true.

7) What is the OEIS A number of this famous sequence?
1,1,2,4,10,26,76,232,764,2620,9496,35696,140152,568504,2390480
A000085

8)Find the operators in N and N^(-1) annihilated by w(n)
w(n+2)-w(n+1)-(n+1)*w(n)=0
N^2-N-(n+1)

9)?SumTools[Hypergeometric[ZeilbergerRecurrence]]
Look up the syntax of ZeilbergerRecurrence
ZeilbergerRecurrence(T, n, k, f, l..u)
where 
T is hypergeoemtric term of n and k
n is name
k is name
f is name of recurrence function
l..u is range for k