#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

#1. 
#(i).Is this in the OEIS (Including the 3 at the beginning)
[seq(nops(KiTours(2,i))/2,i=2..10)];
[3,4,8,16,32,64,128,256,512]
#A198633

#(ii). Without the beginning 3, OEIS?
#very simple,A000079, powers of 2

#(iii). Is the King Tours mentioned in the OEIS in Our meaning?
#nope. It is not

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#.2
#i.How many members of Cn(n) are there with an EVEN number of 1's?
#Cn(1) = 0
#Cn(2) = 2C2 =1
#Cn(3) = 3C2 = 3
#Cn(4) = 4C2+4C4 =
#Cn(n) = nC2+nC4+...+nCi where i=2a<n (a as an integer)
#so,the answer is  
#Cn(n):=add(binomial(n,2i),i=1..n/2)

#ii. How many are there with a number of 1's that is divisible by 4
#By same Tactic
add(binomial(n,4i),i=1..n/4)

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#.3
#i. list of the neighbors of [1,1,1,1,1] in Bn(5)
Bn(5);
[{2, 3, 5, 9, 17}, {1, 4, 6, 10, 18}, {1, 4, 7, 11, 19}, 

  {2, 3, 8, 12, 20}, {1, 6, 7, 13, 21}, {2, 5, 8, 14, 22}, 

  {3, 5, 8, 15, 23}, {4, 6, 7, 16, 24}, {1, 10, 11, 13, 25}, 

  {2, 9, 12, 14, 26}, {3, 9, 12, 15, 27}, {4, 10, 11, 16, 28}, 

  {5, 9, 14, 15, 29}, {6, 10, 13, 16, 30}, {7, 11, 13, 16, 31}, 

  {8, 12, 14, 15, 32}, {1, 18, 19, 21, 25}, {2, 17, 20, 22, 26}, 

  {3, 17, 20, 23, 27}, {4, 18, 19, 24, 28}, {5, 17, 22, 23, 29}, 

  {6, 18, 21, 24, 30}, {7, 19, 21, 24, 31}, {8, 20, 22, 23, 32}, 

  {9, 17, 26, 27, 29}, {10, 18, 25, 28, 30}, 

  {11, 19, 25, 28, 31}, {12, 20, 26, 27, 32}, 

  {13, 21, 25, 30, 31}, {14, 22, 26, 29, 32}, 

  {15, 23, 27, 29, 32}, {16, 24, 28, 30, 31}], [[0, 0, 0, 0, 0], 

  [0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 0, 0, 1, 1], 

  [0, 0, 1, 0, 0], [0, 0, 1, 0, 1], [0, 0, 1, 1, 0], 

  [0, 0, 1, 1, 1], [0, 1, 0, 0, 0], [0, 1, 0, 0, 1], 

  [0, 1, 0, 1, 0], [0, 1, 0, 1, 1], [0, 1, 1, 0, 0], 

  [0, 1, 1, 0, 1], [0, 1, 1, 1, 0], [0, 1, 1, 1, 1], 

  [1, 0, 0, 0, 0], [1, 0, 0, 0, 1], [1, 0, 0, 1, 0], 

  [1, 0, 0, 1, 1], [1, 0, 1, 0, 0], [1, 0, 1, 0, 1], 

  [1, 0, 1, 1, 0], [1, 0, 1, 1, 1], [1, 1, 0, 0, 0], 

  [1, 1, 0, 0, 1], [1, 1, 0, 1, 0], [1, 1, 0, 1, 1], 

  [1, 1, 1, 0, 0], [1, 1, 1, 0, 1], [1, 1, 1, 1, 0], 

  [1, 1, 1, 1, 1]]

#but without the explict Bn(5) we can just think that [1,1,1,1,1] will have [1,1,1,1,0] where 0 can be anywhere. so 5.
#meaning number of neighbors will be same as number of 1s
#So, 5

#ii. how many neighbors does [1,1,1,...] (<- # of a = 10^100 times) in Bn(10^1000) 
#answer: so by same principle, 10^100

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

#4. Find this in OEIS

DiagSeq3(1/(2*x*y*z - x - y - z + 1), x, y, z, 50)

[4, 46, 652, 10186, 168304, 2884456, 50723824, 909192538, 16538659384, 304391739796,...]

#A318109, a(n) = Sum_{k=0..n} (3*n-2*k)!/((n-k)!^3*k!)*(-2)^k.


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

#5.Find DiagSeq3(1/(4*x*y*z - x - y - z + 1), x, y, z, 50) in the oeis

DiagSeq3(1/(4*x*y*z - x - y - z + 1), x, y, z, 50)
[2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932, 38165260, 278415920, 2046924400, 15148345760, 112738423360, 843126957056, 6332299624282, 47737325577620, 361077477684436, 2739270870994736, 20836827035351596, 158883473753259752, 1214171997616258240, 9297206748439774976, 71321835387812803696, 548057409594239814752, 4217997789071070495904, 32509700414484268142720, 250900149362426273826112, 1938786076426160669163392, 14999045536109527518717184, 116163417566507388158025728, 900570078611885122763226970, 6988453515115800846190860404, 54279461366450237575422692644, 421947502214738208737955062960, 3282684865686445066146128050420, 25558123272720374313421408539800, 199131669075939966139717964040160, 1552555975265232644795764435458560, 12112531569476817784029642094074412, 94556099919171802290073846102339064, 738581442841190189324694396536798440, 5772309056432320837068122263930496672, 45136933090913949942301253626857273472, 353130710702357016901082002666296818432, 2764077776735049091680416205763261494784, 21645485694627072265177137089890356961280, 169581416463282016885583341313357923169392, 1329153351023643434414727317328867397924832, 10421983249673701266967875106708685038977760]
#A000172,Franel number a(n) = Sum_{k = 0..n} binomial(n,k)^3.

#The main reason of this seq in OEIS is the Franel number not the diagonal Sequence, however,
#it is mentioned in the comments that it is cross-referenced to 
#Diagonal of rational functions 1/(1 - x*y - y*z - x*z - 2*x*y*z), 1/(1 - x - y - z + 4*x*y*z), 1/(1 + y + z + x*y + y*z + x*z + 2*x*y*z), 1/(1 + x + y + z + 2*(x*y + y*z + x*z) + 4*x*y*z). - Gheorghe Coserea, Jul 04 2018