read `Hardin.txt`:
read `FindRec.txt`:
read `SMAZ.txt`:
K:=40:
L:=SeqB(2,K):
print(`The first`, K, `terms starting at n=1 of the sequence enumerating 6 by 2*n balanced 0-1 matrices, i.e. where every row and every column has the same number 0s and 1s are`):
print(`In other words OEIS sequence  A2896 :`):
print(``):
lprint(L):
print(``):

ope:=Findrec(L,n,N,10):



print(` The sequence, let's call it a(n), enumerating 4 by 2*n balanced 0-1 natrices probably satisfies the empirically derived linear recurrence`):
print(``):
print(add(coeff(ope,N,i)*a(n+i),i=0..degree(ope,N))=0):
print(``):
print(`and in Maple notation`):
print(``):
lprint(add(coeff(ope,N,i)*a(n+i),i=0..degree(ope,N))=0):
print(``):
print(`This took`, time(), `seconds `):
print(``):
t0:=time():
print(``):
print(`Just for fun here is a(1000)`):
print(``):
lprint(SeqFromRec(ope,n,N,[op(1..degree(ope,N),L)],1000)[1000]):
print(``):
print(`This took`, time()-t0, ` additional seconds`):
print(``):
print(`--------------------`):
print(``):
t0:=time():
print(`Now let's do it fully rigorously, using the multivariable Almkvist-Zeilberger algorithm`):

H:=x[1]*x[2]+x[1]*x[3]+x[1]*x[4]+ x[2]*x[3]+x[2]*x[4]+x[3]*x[4]:
H:=H^2/mul(x[i],i=1..4):

ope1:=MAZ(1,1/mul(x[i],i=1..4),H,[seq(x[i],i=1..4)],n,N,{},[y[1],y[2],y[3]],z1)[1]:


lprint(`The empirically derived operator annihilating the sequence is`, ope):
print(``):
lprint(`The rigorously derived operator annihilating the sequence is`, ope1):
print(``):

if ope=ope1 then
 print(`They are indeed the same!`):
else
 print(`Too bad, they are not the same`):
fi:

print(`This took`, time()-t0, `seconds. `):
print(``):