#OK to post homework
#TaerimKim,9/10/2020,Assignment 1

#1. 
> diff(x^5, x);
                                 4

                              5 x 
;
> Int(x^2/sqrt(-x^3 + 1), x);
                       /                  

                      |         2         

                      |        x          

                      |  -------------- dx

                      |           (1/2)   

                     /   /  3    \        

                         \-x  + 1/        
;
> 
;
> evalf(tan(Pi/5));
                          0.7265425281

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

#3-1.
L1 := [Mercury, Venus, Earth];
                 L1 := [Mercury, Venus, Earth]
L2 := [op(L1), Mars];
              L2 := [Mercury, Venus, Earth, Mars]
#3-2: Intuitively, negative steps is not possible since we want only positive step .Or otherwise, we will get infinite regression. So we put empty set to prevent infinite regression.
#3-3: (0,0) to (0,0) is still considered a move but normal move. So we still assign singleton step to count as # of steps. 

#4.
>Count := proc(m, n) local W1, W2, w1, w2; option remember; if m < 0 or n < 0 then RETURN({}); end if; if m = 0 and n = 0 then RETURN({[]}); end if; W1 := Count(m - 1, n - 1); W2 := Count(m - 2, n - 1); {seq([op(w1), 1], w1 in W1), seq([op(w2), 2], w2 in W2)}; end proc;
>F := proc(n) local a; option remember; a := n; do print(Count(n, a))*a--; until a = 0; end proc;
>F(6);
                      {[1, 1, 1, 1, 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, 2, 2], [1, 2, 1, 2], [1, 2, 2, 1], [2, 1, 1, 2], 

     [2, 1, 2, 1], [2, 2, 1, 1]}
                          {[2, 2, 2]}
                               {}
                               {}
##I got what I have calculated with hand but also included empty set as well.. But this is the best I can do!
 

#5.
>ICount := proc(m, n) local W1, W2, w1, w2; option remember; if m < 0 or n < 0 then RETURN(0); end if; if m = 0 and n = 0 then RETURN(1); end if; W1 := ICount(m - 1, n - 1); W2 := ICount(m - 2, n - 1); W1 + W2; end proc
>NuF := proc(n) local a, result; a := n; result := 0; do result := result + ICount(n, a); a--; until a = 0; print(result); end proc
##Caculating NuF(1000)
>NuF(1000);
 70330367711422815821835254877183549770181269836358732742604905087154537118196933579742249494562611733487750449241765991088186363265450223647106012053374121273867339111198139373125598767690091902245245323403501