#OK to post homework
#Ariana Yousafzai, 9/20/2020, Assignment 1


#Question 1


diff(x*y^9,x);
distance((3, 7, 42),(9,9,9); 
int(17*y^3, y); 


#Question 2


F(6) = {[1,1,1,1,1,1],[2,1,1,1,1],[1,2,1,1,1],[1,1,2,1,1],[1,1,1,2,1],[1,1,1,1,2],[2,2,1,1],[2,1,2,1],[2,1,1,2],[1,1,2,2],[1,2,1,2],[1,2,2,1]}


#Question 3


L1 := [Mercury, Venus, Earth]; 
L2 := [op(L1), Mars]; 


We initialize WALKS(m,n) ) with either m or n or both negative to be the Empty set {} because if either value in negative, they exist outside the Manhattan “lattice,” and it is thereby impossible to create a path from one point to the other. 


We initialize WALKS(0,0) to be the singleton set {[]} because there is no path that exists that connects one point to the other since the function indicates that the start and end point is the same. 


#Question 4


F:=proc(n) local X1,X2,x1,x2:
option remember:


if n<0 then
 RETURN({}):
fi:


if n=0 then
 RETURN({[]}):
fi:


X1:=F(n-1):
X2:=F(n-2):


{seq([op(x1),1],x1 in X1), seq([op(x2),2],x2 in X2)  }:


end:


F(6) = {[2, 2, 2], [1, 1, 2, 2], [1, 2, 1, 2], [1, 2, 2, 1],  [2, 1, 1, 2], [2, 1, 2, 1], [2, 2, 1, 1], [1, 1, 1, 1, 2], [1, 1, 1, 2, 1], [1, 1, 2, 1, 1], [1, 2, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]}


#This matches the F(6) determined by hand. 


#Question 5


#NuF(n): inputs a non-negative integer n and outputs the numbers of lists in {1,2} that add up to n
NuF:=proc(n) local X1,X2,x1,x2:
option remember:


if n<0 then RETURN(0): fi:
if n=0 then RETURN(1): fi:


F1:=NuF(n-1):
F2:=NuF(n-2):


X1+X2:
end:


NuF(1000) = 70330367711422815821835254877183549770181269836358732742604905087154537118196933579742249494562611733487750449241765991088186363265450223647106012053374121273867339111198139373125598767690091902245245323403501


# nops(F(1000)) commands Maple to compute the number of elements of F(1000), which comes out to an extremely large number which is overwhelming for the program to handle.