read `Motzkin.txt`:

print(`Here are examples with restrictions on peak heights, valley heights, and runs`):
print(``):
print(``):
print(``):

print(`Let a(n) be the number of Motzkin paths of length n`):
L:=SeqABCDEr({},{},{},{},{},r,100):
eq:= Aeq(L,x,P):
print(`The generating function P(x) of the sequence a(n) satisfies the algebraic equation`):
if eq=FAIL then 
  print(eq):
  print(`Cannot find the equation from the first 101 terms of the sequence`):
print(eq=0):
else 
  print(eq=0):
fi:
print(``):
print(``):
print(`Here are the terms a(n) from n=0 to n=100`):
print(``):
print(L):

print(``): print(``): print(`----------------------------`): print(``): print(``):



print(`Let a(n) be the number of Motzkin paths of length n with the following restrictions`):
print(`No peak height can belong to {2r+1} and no upward run can belong to {2r+1}`):
L:=SeqABCDEr({2*r+1},{},{2*r+1},{},{},r,100):
eq:= Aeq(L,x,P):
print(`The generating function P(x) of the sequence a(n) satisfies the algebraic equation`):
if eq=FAIL then 
  print(eq):
  print(`Cannot find the equation from the first 101 terms of the sequence`):
print(eq=0):
else 
  print(eq=0):
fi:
print(``):
print(``):
print(`Here are the terms a(n) from n=0 to n=100`):
print(``):
print(L):


print(``): print(``): print(`----------------------------`): print(``): print(``):

print(`Let a(n) be the number of Motzkin paths of length n with the following restrictions`):
print(`No valley height can belong to {2r+1} and no upward run can belong to {2r+1}`):
L:=SeqABCDEr({},{2*r+1},{2*r+1},{},{},r,100):
eq:= Aeq(L,x,P):
print(`The generating function P(x) of the sequence a(n) satisfies the algebraic equation`):
if eq=FAIL then 
  print(eq):
  print(`Cannot find the equation from the first 101 terms of the sequence`):
print(eq=0):
else 
  print(eq=0):
fi:
print(``):
print(``):
print(`Here are the terms a(n) from n=0 to n=100`):
print(``):
print(L):


print(``): print(``): print(`----------------------------`): print(``): print(``):

print(`Let a(n) be the number of Motzkin paths of length n with the following restrictions`):
print(`No valley height can belong to {2r+1} and no flat run can belong to {2r+1}`):
L:=SeqABCDEr({},{2*r+1},{},{},{2*r+1},r,100):
eq:= Aeq(L,x,P):
print(`The generating function P(x) of the sequence a(n) satisfies the algebraic equation`):
if eq=FAIL then 
  print(eq):
  print(`Cannot find the equation from the first 101 terms of the sequence`):
print(eq=0):
else 
  print(eq=0):
fi:
print(``):
print(``):
print(`Here are the terms a(n) from n=0 to n=100`):
print(``):
print(L):

print(``): print(``): print(`----------------------------`): print(``): print(``):

print(`Let a(n) be the number of Motzkin paths of length n with the following restrictions`):
print(`No peak height, valley height, upward-run, downward-run, or flat-run can belong to {2r+1}`):
L:=SeqABCDEr({2*r+1},{2*r+1},{2*r+1},{2*r+1},{2*r+1},r,100):
eq:= Aeq(L,x,P):
print(`The generating function P(x) of the sequence a(n) satisfies the algebraic equation`):
if eq=FAIL then 
  print(eq):
  print(`Cannot find the equation from the first 101 terms of the sequence`):
print(eq=0):
else 
  print(eq=0):
fi:
print(``):
print(``):
print(`Here are the terms a(n) from n=0 to n=100`):
print(``):
print(L):


print(``): print(``): print(`----------------------------`): print(``): print(``):

print(`Let a(n) be the number of Motzkin paths of length n with the following restrictions`):
print(`No peak height can belong to {3*r+1} and no downward-run can belong to {3*r+2}`):
L:=SeqABCDEr({3*r+1},{},{},{3*r+2},{},r,100):
eq:= Aeq(L,x,P):
print(`The generating function P(x) of the sequence a(n) satisfies the algebraic equation`):
if eq=FAIL then 
  print(eq):
  print(`Cannot find the equation from the first 101 terms of the sequence`):
print(eq=0):
else 
  print(eq=0):
fi:
print(``):
print(``):
print(`Here are the terms a(n) from n=0 to n=100`):
print(``):
print(L):