#ATTENDANCE QUIZ FOR LECTURE 14 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p24
#with an attachment called
#p24FirstLast.txt
#(e.g. p24DoronZeilberger.txt)

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


THE NUMBER OF ATTENDANCE QUESTIONS WERE:10

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



Question 1:

Write down all the integer partitions of 7 BY HAND.


Answer 1:

[7]
[4, 3], [5, 2], [6, 1], 
[3, 2, 2], [3, 3, 1], [4, 2, 1], [5, 1, 1]
[2, 2, 2, 1], [3, 2, 1, 1], [4, 1, 1, 1]
[2, 2, 1, 1, 1], [3, 1, 1, 1, 1]
[2, 1, 1, 1, 1, 1]
[1, 1, 1, 1, 1, 1, 1]



Question 2:

Write down the integer partition of 7 into distinct parts and odd parts (should be of equal size - Euler's theorem)


Answer 2:

PnD(7,1)
{[7], [4, 3], [5, 2], [6, 1], [4, 2, 1]}


PnC(7,{1},2)
{[7], [3, 3, 1], [5, 1, 1], [3, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]}



Question 3:

What is the A-number of this sequence ?

1,2,3,5,7,11,15,22,30,42,56,77

Answer 3:

A000041		
a(n) is the number of partitions of n (the partition numbers)




Question 4:

Who wrote the classic book on the theory of partitions? What is his birthdate?


Answer 4:

George E. Andrews wrote the classic book on the theory of Partitions. 

His birthday is December 04, 1938.




Question 5:

What is the A-number of this sequence?

1,1,2,2,3,4,5,6,8,10,12,15,18,22,27,32...


Answer 5:

A000009		
Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n 
into distinct parts; number of partitions of n into odd parts.




Question 6:

What is the A-number of this sequence?

1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,17,19,23...

Answer 6:

A000607	
Number of partitions of n into prime parts.



Question 7:

seq(nops(PnC(i,{1,4},5)),i=..20)

seq(nops(PnC(i,{1,4},5)),i=..20)


Who's theorem is it that the above two sequences are the same?


Answer 7:

Srinivasa Ramanugan



Question 8:

6,27,98,315,913,2462,6237,15035...

Is it in the OEIS? If so, what is its A-number?

Answer 8:

Yes it is in the OEIS.

A071734		
a(n) = p(5n+4)/5 where p(k) denotes the k-th partition number.




Question 9:

(i)
Does there exist a k between 0 and 6 such that seq(pn(7*n+k)/7,n=1..20) is an integer sequence?

(ii)
Does there exist a k between 0 and 10 such that seq(pn(11*n+k)/11,n=1..20) is an integer sequence?


Answer 9:

(i)

Yes, k=5 produces an integer sequence.

seq([seq(pn(7*n+k)/7,n=1..20)],k=0..6)[6]

[11, 70, 348, 1449, 5334, 17822, 55165, 160215, 441105, 1159752, 2929465, 7142275, 16873472, 38749850, 86737678, 189672868, 405991500, 852077072, 1756048833, 3558408287]



(ii)

Yes, k=7 produces an integer sequence.

seq([seq(pn(11*n+k)/11,n=1..20)],k=0..10)[7]

[27, 338, 2835, 18566, 101955, 490253, 2121679, 8424520, 31120519, 108082568, 355805845, 1117485621, 3366123200, 9767105406, 27398618368, 74534264393, 197147918679, 508189847045, 1279140518117, 3149375120229]



Question 10:

Why should Dr.Z from now on make you watch the lectures before the recitations?


Answer 10:

I think this would be more beneficial since students would actually have questions corresponding
To the lecture material. 

It would be a good idea to have the attendance quiz due the day before the recitation so everyone actually 
Watches the lecture and can then come to recitation if they have any questions.