#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 3,4,8,16,32,64,128,256,512,... in the OEIS? What is the A-number?
(ii) What is the A-number of the sequence without the leading 3?
(iii) Is the king tours mentioned in the OEIS in our meaning?

ANSWER:

(i) Yes, its A-number is A198633: Total number of round trips, each of length 2*n on the graph P_3 (o-o-o).
(ii) Yes, its A-number is A020707: Pisot sequences E(4,8), L(4,8), P(4,8), T(4,8).
(iii) Yes, for example A-number A140519 is the sequence of undirected Hamiltonian cycles on the n X n king graph.


QUESTION #2:

(i) How many members of C_n(n) are there with an even number of 1s.
(ii) How many are there with a number of 1s that is divisible by 4.

ANSWER:

(i) In general, if n is a natural number then we have two cases:
if n is even then we have  binomial(n,2) + binomial(n,4) + binomial(n,6) + ... + binomial(n,n)
if n is odd then we have binomial(n,2) + binomial(n,4) + binomial(n,6) + ... + binomial(n,n-1)

(ii) In general, there are sum(binomial(n,4k), k=1..n/4) members of C_n(n) with a number of 1s that is divisible by 4.


QUESTION #3:

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

ANSWER:

(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] - 5 neighbors
(ii) A neighbor of [1,1,1,1,...,1] is found by changing one of the 1s to a 0 and there are 10^100 ways to do this. So, it has 10^100 neighbors.


QUESTION #4:

Is the sequence 4,46,652,10186,168304,2884456,50723824 in the OEIS? What is its A-number?

ANSWER:

A close sequence which starts with a 1 is A318109: a(n) = Sum_{k=0..n} (3*n-2*k)!/((n-k)!^3*k!)*(-2)^k.


QUESTION #5:

Is the sequence 2,10,56,346,2252,15184,104960,739162,5280932 in the OEIS? What is its A-number? Is that the main reason why its there? Is the fact that it is the diagonal sequence of 1/(1-x-y-z+4*x*y*z) mentioned there?

ANSWER:

It is in the OEIS, with A-number A000172. The main reason it is in there is because it represents the sequence of Franel numbers: a(n) = Sum_{k = 0..n} binomial(n,k)^3. It is mentioned that it is the diagonal sequence of 1/(1-x-y-z+4*x*y*z) by Gheorghe Coserea.