#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 form to explicit form.

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


Answer 1:


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



Question 2:


Find the above sequence by hand for f(5) and f(6)


Answer 2:

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



Question 3:


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


Answer 3:

#MAPLE timed out - took tooo long

F:=proc(n) local i,L: L:=[1,-2]: for i from 1 to n-1 do L:=[L[2],3*L[2]-4*L[1]] od: L[2]: end:
F(1000000)



Question 4:

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


Answer 4:


ope:=


Question 5:

Characterize the sequences that satisfy a homogenous recurrence of order 0.


Answer 5:

- It is a constant sequence



Question 6:

Can you prove that d(n)/n!->1/e?


Answer 6:

!n = n! * Sum((-1^i)/I!, i=0..n)

e^x = Sum((x^i)/(i!),i=0..infinity)

Using x=-1

lim x-> infinity (!n/n!) = 1/e


Question 7:

What is the OEIS number of this sequence?
(The sequence of involutions)

Answer 7:

A000085



Question 8:


Find the operators in N and N^-1 annihilated by w(n) (sequence of involutions)


Answer 8:

- N^2 - N - (n-1)

- 1 - N^-1 - (n+1)N^-2



Question 9:


Look up the syntax of the Zeilberger recurrence. 


Answer 9:

with(SumTools[Hypergeometric])
	
Zpair≔Zeilberger(T,n,k,En):