#ATTENDANCE QUIZ FOR LECTURE 9 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p9
#with an attachment called
#p9FirstLast.txt
#(e.g. p9DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Oct. 6, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 5

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

QUESTION #1:

Let a[i] be the ith digit of your RUID. In how many ways can you walk from 0 to 131 using as fundamental steps the set {a[3], a[5], a[9]}

ANSWER:

Based off of my RUID, S = {9, 1, 8}.
This yields the generating function f(x) = 1/(1-x-x^8-x^9).
We can find the the number of ways to walk from 0 to 131 by finding the coefficient to 132nd coefficient of the Taylor series for the function:
GFseq(1/(1-x-x^8-x^9),x,131)

The coefficient a(131) is 157976394384193.

QUESTION #2:

Is the sequence 1, 1, 1, 1, 2, 3, 5, 7, 10, 15, 23, 35, 52, 77, 115, 173, 260, 389, 581, 869, 1302, 1951, 2921, 4371, 6542, 9795, 14667, 21959, 32872, 49209, 73671, 110297, 165128, 247209, 370089, 554057, 829482, 1241819, 1859117, 2783263, 4166802 in the OEIS?

ANSWER:

Searching resulted in 0 hits. Not in OEIS.

QUESTION #3:

Let f(x) = 1/(1-4x-x^6).
(i) Find the coeff of x^100 in the taylor expansion of f(x).
(ii) Find the coeff of x^101 in the taylor expansion of f(x).
How far is a(101)/a(100) from the real root of 1-4x-x^6.

ANSWER:

Using, GFseq(1/(1-4x-x^6),x,101):
(i) a(100) = 1644594257296515568488059327938971072084521685469936722570496
(ii) a(101) = 6579981121576468112367189280234473995254094130576501178758144.

a(101)/a(100) = 4.000975373
fsolve(1-4x-x^6) = 0.2499390541
4.000975373 - 0.2499390541 = 3.751036319.

QUESTION #4:

In how many ways can a chess king walk from 1 corner of the chess board to the opposite corner?

ANSWER:

We want the king to walk from [0,0] to [7,7].
The valid moves for the king are the set S={[0,1],[1,0],[1,1]).
Hence, we have the generating function:
f(x,y) = 1/(1-Sum(x^s[1]*x^s[2], s in S)).
The number of walks is given by a(7,7) of the Taylor expansion for the generating function.
We can find this number by executing:
DiagWalks2D({[0,1],[1,0],[1,1]},7);

Answer: 48639

QUESTION #5:
f:=normal(1/(1-x*(1-x)-y*(1-y)))
A:= (m,n) -> coeff(taylor(coeff(taylor(f,x=0,m+1),x,m),y=0,n+1),y,n);
seq(A(i, i), i = 0 .. 20) = 1, 2, 2, 4, 16, 40, 76, 184, 538, 1372, 3200, 8096, 21724, 55928, 140072, 360464, 945728, 2451128, 6302356, 16357064, 42740780
   
Is seq(A(i,i),I=0..20) in the OEIS?

ANSWER:

This sequence returned 0 hits in OEIS. It's not in OEIS.