#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

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Attendance Question 1:
(i) Is the sequence [3,4,8,16,32,64,128,256,512] in the OEIS? What is the A number?
(ii) Without the 3, what is the A number?
(iii) Is the King Tours mentioned in the OEIS in OUR meaning?

i) There are two sequences in the OEIS that contain this sequence: A198633 and A244750. 
ii) A79: Powers of 2
iii) Yes, the closest I could find is A140519 which is the number of undirected hamiltonian cycles on the nxn king graph (although we aim to look at the kxn king graph)

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Attendance 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 1s that is divisible by 4?

i) Exactly half of the members of Cn(n) have an even number of 1's (i.e 2^{n-1})
ii) [1,1,1,2,6,16,36,72,136,256,496...]. Doesn't seem to have a closed form solution. 
Number of 1s divisible by 4 = Sum_{k = 0..floor(n/4)} binomial(n, 4*k).
The generating function is (1-x)^3/((1-x)^4-x^4).

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Attendance Question 3:

i) List the Neighbors of [1,1,1,1,1] in Bn(5)
ii) How many neighbors does [1,1,1,...,1] (1 repeated 10^100 times) in Bn(10^1000) have?


i) {[0,1,1,1,1],[1,0,1,1,1],[1,1,0,1,1],[1,1,1,0,1],[1,1,1,1,0]}
ii) There are 10^100 neighbors of the specified vertex.

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Attendance Question 4:

Is this sequence in the OEIS?

[4,46,652,10186,168304,...]

Yes, it is A318109.


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Attendance Question 5:

What is the A-number of this sequence?
Is that the MAIN Reason why it is there?
Diagonal sequence of 1/(1-x-y-z+4*x*y*z) mentioned?

[2,10,56,346,2252,15184,104960,...]

The A-number is A172. It is not the main reason it is there (the main reason is is they are the Franel numbers a(n) = Sum_{k=0..n} binomial(n,k)^3).
However, the diagonal sequence of 1/(1-x-y-z+4*x*y*z) is mentioned in the comments.