#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:4

#attendance question 1.

#Let a[i] be the i-th digit of your RUID (if its 0, make it 1)
#In how many ways can you walk from 0 to 131 using as fundamental steps
#the set {a[3],a[5],a[9]}?

#answer: given RUID as 197005262,
#set A:={7,1,2}
#we use the theorem to get the generating function 
#f=1/(1-x-x^2-x^7)
v := normal(1/(-x^7 - x^2 - x + 1));
                                   1       
                     v := - ---------------
                             7    2        
                            x  + x  + x - 1
coeff(taylor(v, x = 0, 132), x, 131);
                 26740756499377665109292495248



################################

#attendance question 2.
#f:=normal(1/(1-(x+x^4)/(1-x^5))):
#coeff(taylor(f,x=0,102),x,101):
#is this sequence in OEIS?

#Answer:
seq(coeff(taylor(f,x=0,102),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, 6238103, 9339039, 13981419, 20931484, 31336389, 46913531, 70233989, 105146892, 157414765, 235664685, 352812205, 528193086, 790754743, 1183834193, 1772311083, 2653316374, 3972264203, 5946853139, 8902998415, 13328625872, 19954206449, 29873323791, 44723175345, 66954799632, 100237631953, 150065162193, 224661661329, 336339636306, 503532067891, 753834862037, 1128561685559, 1689562983194, 2529434687391, 3786801617319, 5669198164915, 8487322833668, 12706320504253, 19022556808963, 28478556591197, 42635077589780, 63828720927701, 95557598240917, 143058711641077, 214172345822054, 320636144339535, 480022463508153, 718638773390147, 1075869830853278, 1610678321014867, 2411336928862555, 3609998165760855, 5404506770004280, 8091054921872425, 12113070171749847, 18134405266373257, 27148910202138392, 40644471894015097, 60848596987637369, 91096072425760473, 136379387894272122, 204172769990425611

#but it doesnot come out in OEIS, The similiest one is A286227, Number of compositions (ordered partitions) of n.
##################################


#attendance question 3.
#let f(x)=1/((1-4x-x^6):
#(i). Find the coefficient of x^100 in the taylor expansion of f(x)
#(ii). Find the coefficient of x^101 in the taylor expansion of f(x)
#how far is the a(100)/a(101) from the real root?

#answer:
#i. 
g := 1/(-x^6 - 4*x + 1);
coeff(taylor(g, x = 0, 101), x, 100);
 1644594257296515568488059327938971072084521685469936722570496
#ii,
coeff(taylor(g, x = 0, 102), x, 101);
 6579981121576468112367189280234473995254094130576501178758144

evalf(1644594257296515568488059327938971072084521685469936722570496/6579981121576468112367189280234473995254094130576501178758144);
                          0.2499390541

denom(g);
                           6          
                          x  + 4 x - 1
fsolve(%);
                   -1.364656529, 0.2499390541

#here we can see that a(100)/a(101) is the real root of the function which agrees with the fundamental theorem that reciprocal of a(101)/a(100) will be same as the real root.

#####################################

#attendance question 4.
#in How many ways can a chess king walk from one corner of the chess board
#to the opposite corner?

#answer: noting that king is alway moving forward, we are in where S={[1,0],[0,1],[1,1]} from
#for a(8,8)
#generating function is
#f=1/(1-x-y-xy)
coeff(taylor(coeff(taylor(%,x=0,9),x,8),y=0,9),y,8)