#ATTENDANCE QUIZ FOR LECTURE 17 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p17
#with an attachment called
#p17FirstLast.txt
#(e.g. p17DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Nov. 3, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 5

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


Question 1:
   (i) Is this ([3,4,8,16,32,64...]) in the OEIS (including the 3 at the beginning)? What is the A number? 
   (ii) Without 3 it is, what is the A number?
   (iii) Is the King Tours mentioned in the OEIS in our meaning? 

ANSWER:
(i) A198633
(ii) A179
(iii) The Kings Tour is not mentioned in the OEIS in our meaning. 

Question 2:
   (i) How many members of Cn(n) are there with an even number of 1's?
   (ii) How many are there with a number of 1's that is divisible by 4? 

ANSWER:
(i) There are (n C 0) + (n C 2) + ... + (n C n) if n is even and (n C 0) + (n C 2) + .... + (n C n-1) if n is odd. 
(ii) There are (n C 0) + (n C 4) + ... + (n C n) if n is divisible by 4.
     There are (n C 0) + (n C 4) + ... + (n C n-1) if n mod 4 is 1. 
     There are (n C 0) + (n C 4) + ... + (n C n-2) if n mod 4 is 2. 
     There are (n C 0) + (n C 4) + ... + (n C n-3) if n mod 4 is 3.   
   
Question 3:
   (i) List of the neighbors of [1,1,1,1,1] in Bn(5)
   (ii) How many neighbors does [1,1,1,1...., 1] (1 repeated to the 10^1000)
        in Bn(10^1000) have? 

ANSWER:
(i) [1,1,1,1,0], [1,1,1,0,1], [1,1,0,1,1], [1,0,1,1,1], [0,1,1,1,1] are the neighbors.
(ii) [1,1,1,1...., 1] (1 repeated to the 10^1000) should have 10^1000 neighbors.

Question 4: 
Is this sequence in the OEIS? [4,46,652,10186, 168304...]

ANSWER: A318109

Question 5. 
What is the A number of this sequence [2,10,56, 346, 2252 ...]? Is that the main reason why it is in 
the OEIS? Is the fact it is the diagonal sequence of 1/(1-x-y-z+4xyz) mentioned? 

ANSWER: A172. The main reason why it is in the OEIS is that it is the Franel number a(n) = Sum_{k = 0..n} binomial(n,k)^3.
The fact that it is the diagonal sequence of 1/(1-x-y-z+4xyz) is later mentioned.