#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



#ATTENDANCE Q. #1 for LECTURE 17
#Is this KiTours seq in the OEIS? What is the A number? (i)
#Without the beginning 3? (ii)
#Is the King Tours mentioned in our meaning?

#ANSWER to Q. #1:	

# (i) Yes, it is:   A198633	"Total number of round trips, each of length 2*n on the graph P_3"
# (ii) Yes, powers of 2:  A000079
# (iii) No, "king" isn't mentioned, but it talks about paths likely using fundamental steps of 1 in any direction,
# which is what our King does (can walk in any direction with unit step)




#ATTENDANCE Q. #2 for LECTURE 17
#How many members of Cn(n) have an even number of 1s? How many have a number of 1s that is divisible by 4?

#ANSWER to Q. #2:

#(i) 2^(n-1) members of Cn(n) have an even number of 1s
#(ii) there are 1,1,1,1,2,6,16,36,72... or the sum of every 4th entry of row n in Pascal's triangle, starting
# at binomial(n,0) (from OEIS)
#	It has generating function (1-x)^3/((1-x)^4-x^4)




#ATTENDANCE Q. #3 for LECTURE 17
#List the neighbors of [1,1,1,1,1] in Bn(5)
#How many neighbors does [1,1,1,...,1,1,1] (1 repeated 10^100 times) in Bn(10^100) have?

#ANSWER to Q. #3:

#(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) It has 10^100 neighbors




#ATTENDANCE Q. #4 for LECTURE 17
#Is that sequence in the OEIS?

#ANSWER to Q. #4:

#Yes, its A number is A318109 ("a(n) = Sum_{k=0..n} (3*n-2*k)!/((n-k)!^3*k!)*(-2)^k")




#ATTENDANCE Q. #5 for LECTURE 17
#A number of the sequence with 4 instead of 2? Is that the main reason why it is there?
#Is the fact that it is the diagonal sequence of 1/(1-x-y-z+4*x*y*z) mentioned there?

#ANSWER to Q. #5:

#A number = A000172
#Its main reason is that of being the sequence of "Franel" numbers and there are many other definitions below
#Yes, it is mentioned that it is the diagonal of rational functions such as the ones we went over