#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:

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


Question 1:

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

Answer 1:
RUID: 187006241
The corresponding set of steps is {7,1,1}

normal(1/(1-x^7-x^1-x^1))
	-1/(x^7+2*x-1)
coeff(taylor(%,x=0,132),x,131)
	6910387727064368901537947629720143683104



Question 2:
Is the above sequence in OEIS?

Answer 2:

f:=normal(1/(1-(x+x^4)/(1-x^5))):

seq(coeff(taylor(f,x=0,41),x,i),i=0..40)

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

No the sequence is not in the OEIS.



Question 3:

Let f(x)=1/(1-4x-x^6)
Find the coefficients of x^100 and x^101.
How far is x^[100]/x^[101] from the real root of (1-4x-x^6)?


Answer 3:

Reverse direction. (In class, we did L[101]/L[100])
L[100]/L[101]
6422630202975873673507693414568221301177830793440184959945/25696785270258055757625926999046423001320651335467761290164

evalf(%)
.2499390541

fsolve(denom(f))
-1.364656529, .2499390541

It is one of the solutions of the denominator. 






Question 4:

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


Answer 4:

f:=normal(1/(1-x/(1-x) -y/(1-y)));
f := (-1+x)*(-1+y)/(3*x*y-2*x-2*y+1)

A:=(m,n)-> coeff(taylor(coeff(taylor(f,x=0,m+1),x,m),y=0,n+1),y,n);

A := proc (m, n) options operator, arrow; coeff(taylor(coeff(taylor(f, x = 0, m+1), x, m), y = 0, n+1), y, n) end proc

A(7,7)
470010







Question 5:

Is the sequence mentioned in the lecture in the OEIS?


Answer 5:


f:=normal(1/(1-x/(1-x) -y/(1-y)));
f := (-1+x)*(-1+y)/(3*x*y-2*x-2*y+1)


A:=(m,n)-> coeff(taylor(coeff(taylor(f,x=0,m+1),x,m),y=0,n+1),y,n);
A := proc (m, n) options operator, arrow; coeff(taylor(coeff(taylor(f, x = 0, m+1), x, m), y = 0, n+1), y, n) end proc


seq(A(i,i),i=0..20);
1, 2, 14, 106, 838, 6802, 56190, 470010, 3968310, 33747490, 288654574, 2480593546, 21400729382, 185239360178, 1607913963614, 13991107041306, 122002082809110, 1065855419418690, 9327252391907790, 81744134786314410, 717367363052796678


Yes the sequence is in the OEIS.
A051708