Everything seems to be working properly (or you expect other answers):

restart;
L:= [1,2,3,4,5,6,7,8];
Statistics:-Quartile(L,1);
Statistics:-Quartile(L,2);
Statistics:-Quartile(L,3);

                                L := [1, 2, 3, 4, 5, 6, 7, 8]
                                2.4166666666666666666
                                4.4999999999999999999
                                6.5833333333333333333

B:=N->Matrix(N+1, {seq((i,i+1)=i, i=1..N)});

Example of use:
B(5);

restart:
x := 12:
y := 46:
s := 0;
'x' = x:
'y' = y:
'`résultat`' = s: k:=0:
while 0 < y do
if type(y, odd) then s := s + x:
y := y - 1:
else x := 2*x:
y := y/2:
end if;
k:=k+1; L[k] := [x, y, s];
end do;

op~(convert(L,list));

Example of use:

combinat:-partition(4);
             [[1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3], [4]]

You may be interested in this post  https://www.mapleprimes.com/users/Kitonum/posts?page=6

If you add one more parameter (I added  m ), Maple will find a huge number of solutions. Below is found and built the plot for one solution:

Download Equation.mw

Yes, I can confirm that the legend option does not work (I checked in Maple 2018.2). In a sense, this is natural, because legend option is associated with the image of curves, rather than regions. Use  caption  instead, as in the example below:

plots:-inequal(x^2+y^2<=1, x = -1.2..1.2,y=-1.2..1.2, caption = typeset("A region of  ", x^2+y^2<=1, "."), captionfont=[times,bold,16], scaling=constrained);

You can put this text anywhere on the screen. See  ?plot,typesetting

If you need to put legends for curves, you can do as below:

A:=plots:-inequal(x^2+y^2<=1, x = -1.2..1.2,y=-1.2..1.2, color=pink, title = typeset("A region of  ", x^2+y^2<=1, "."), titlefont=[times,bold,18], scaling=constrained):
B:=plots:-implicitplot(x^2+y^2=1, x = -1.2..1.2,y=-1.2..1.2, color=red, thickness=3, legend = typeset("A curve of  ", x^2+y^2=1, "."), legendstyle=[font=[times,14]]):
plots:-display(A,B); 

Do a double loop: in the internal one place what you want to suppress, and in the external one everything else and end with a semicolon as in the example:

restart;
for i from 1 to 3 do
for j from 1 to 1 do    
    A := x;
 end;  
B:=C;
end;

It is better to always end a loop with the colon, and if you need to see something in the body of the loop as it is executed, then use  print  as in the example:

for i from 1 to 3 do    
    A := x:
   B := C;
print(%);
end:

 eq:=s=1+2*((tau-t)/T0); 
EQ:=int(f1(t-tau)*(Sum(y[k]*F[k](tau), k = 0 .. M)), tau = t-T0 .. t);
IntegrationTools:-Change(EQ, eq, s);

I do not understand what you are trying to do. Why not just write the following 2 lines and it will be beautiful:

restart;
eq := piecewise(t < 1, sin(t), cos(t));
DocumentTools:-Tabulate(<eq, plot(eq, discont)>, width=50):

Download simpl.mw

Here is a procedure for this:

restart;
MatrixPoly:=proc(nu,M)
local k, Poly, POL; 
for k while k <= M do
Poly[k] := simplify(sum(x^i*GAMMA(nu+1)/(factorial(i)*GAMMA(2*nu)), i = 0 .. k-1));
end do:
POL := <seq(Poly[k], k = 1 .. M)>;
Matrix(M,M, (i,j)->Poly[i]((j-1)/(M-1)));
end proc:

# Examples
MatrixPoly(1,3);
MatrixPoly(1,4);

Edit. Regarding the second problem, it seems I am guessing that you want to create a vector consisting of the values of some function if the argument runs from  a  to  b  through equal steps (total  M  points). If so then

Vec:=(a,b,f,M) -> <seq(f(a+(b-a)*k/(M-1)),k=0..M-1)>:

# Example
Vec(1,2,sin,10)

We can put quotes around function names and omit the square brackets. This will work a little faster:

plot3d('R_ep'(epsilon, t-n, n, 'qN_ep'(epsilon, t-n, n)), n = 1 .. t, epsilon = 10^(-8) .. 10^(-1));

I do not see any bugs in these calculations. Note that  symbolic option works with symbolic expressions. When working with numeric expressions, it is simply ignored. Compare

simplify((p^3)^(1/3),symbolic);
simplify(((-1)^3)^(1/3),symbolic);

                               p
                     1/2+I*sqrt(3)*(1/2)

The second calculation is also correct, as the default Maple works in the complex domain, and returns the principal value of the root (of three values). If you work in the real domain, then use the  surd  function instead of raising to the power 1/3:

surd(-1, 3);

                                 -1

Maple does not simplify the value for (-1)^(1/7)  (in contrast to (-1)^(1/3)), because it is not expressed in radicals, but can be calculated numerically:

simplify((-1)^(1/7));
evalf(%);

                                  (-1)^(1/7)
                    0.9009688678+0.4338837392*I

Maple can also convert   (-1)^(1/7)  to trigonometric form:

convert((-1)^(1/7), trig);
# Or
evalc((-1)^(1/7));

                            cos((1/7)*Pi)+I*sin((1/7)*Pi)
                            cos((1/7)*Pi)+I*sin((1/7)*Pi)

I is protected in Maple (it is the imaginary unit). I replaced I with J :

matrix_inversion_new.mw