> #Attendence Q1:
> #In how many ways can you walk from 0 to 131 using as fundamental steps the set {a[3], a[5], a[9]}? 179009929
;
> #Ans: a[3] = 7, a[5] = 1, a[9] = 9, so {1,7,9}. 
;
> #f(x) := 1/(1-x-x^7-x^9)
;
> #763647434231397
;
> 
;
> #Attendence Q2: 
> #Is this sequence in the OEIS?
;
> f:= normal(1/(1-(x+x^4) /(1-x^5)));
Typesetting:-mprintslash([(f := (x^5-1)/(x^5+x^4+x-1))],[(x^5-1)/(x^5+x^4+x-1)]
)

;
> L:= [seq(coeff(taylor(f, x=0, 41),x,i), i=0..100)]
Typesetting:-mprintslash([(L := [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, O(1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])],[[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, O(1), 0, 0, 0, 0, 0
, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
, 0]])

;
> L
[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, O(1
), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0]

;
> #NO!
;
> 
;
> 
;
> 
;
> #Attendence Q3: 
> #Let f(x)=1/(1-4*x-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)
;
> #(iii) how far is a(101)/a(100) from the real root of 1-4*x-x^6
;
> a101:= [seq(coeff(taylor((1/(1-4*x-x^6)), x=0, 101),x,i) , i=0..100)][101]
Typesetting:-mprintslash([(a101 := 
1644594257296515568488059327938971072084521685469936722570496)],[
1644594257296515568488059327938971072084521685469936722570496])

;
> a100:=[seq(coeff(taylor((1/(1-4*x-x^6)), x=0, 101),x,i) , i=0..100)][100]
Typesetting:-mprintslash([(a100 := 
411048332990455915104492378532366163275381170780171837436480)],[
411048332990455915104492378532366163275381170780171837436480])

;
> a:=evalf(a101/a100)
Typesetting:-mprintslash([(a := 4.000975373)],[4.000975373])

;
> f:= 1/(1-4*x-x^6)
Typesetting:-mprintslash([(f := 1/(-x^6-4*x+1))],[1/(-x^6-4*x+1)])

;
> denom(f)
x^6+4*x-1

;
> fsolve(%)
;
> b:=1/.2499390541
Typesetting:-mprintslash([(b := 4.000975372)],[4.000975372])

;
> abs(a-b)
.1e-8

;
> 
;
> 
;
> 
;
> #Attendence Q4:
> #In how many ways can a chess king walk from one corner of the chess board to the opposite corner?
;
> f:=normal(1/(1-x/(1-x) - y/(1-y)));
Typesetting:-mprintslash([(f := (-1+x)*(-1+y)/(3*x*y-2*x-2*y+1))],[(-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);
Typesetting:-mprintslash([(A := (m, n) -> coeff(taylor(coeff(taylor(f,x = 0,m+1
),x,m),y = 0,n+1),y,n))],[(m, n) -> coeff(taylor(coeff(taylor(f,x = 0,m+1),x,m)
,y = 0,n+1),y,n)])

;
> A(7,7);
470010

;
> 
;
> 
;
> 
;
> #Attendence Q5: in the OEIS?
;
> 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! 
> #A051708		Number of ways to move a chess rook from #the lower left corner to square (n,n), with the rook moving only up or right
;
> 
;