2017-02-10-083524.sagews

def F(n): return sum((n-i+1)*euler_phi(i) for i in range(1,n+1))
for n in range(7):
    F(n)

0 1 3 7 13 23 35

def words(n):
    return Tuples([0,1],n).list()
def mots_equ(n):
    X=[]
    for w in words(n):
        Fact=[]
        for l in range(1,n+1):
            for pos_init in range(n-l+1):
                Fact.append(w[pos_init:pos_init+l])
        Fact_loin=[]
        for u in Fact:
            for v in Fact:
                    if len(u)==len(v) and sum(u)-sum(v) not in [-1,0,1]:
                        Fact_loin.append((u,v))
        Fact_loin
        if len(Fact_loin)==0:
            X.append(w)
    return X
def E(n):
    return len(mots_equ(n))

mots_equ(0)
E(0)

[[]] 1

mots_equ(1)
E(1)

[[0], [1]] 2

mots_equ(2)
E(2)

[[0, 0], [1, 0], [0, 1], [1, 1]] 4

mots_equ(3)
E(3)

[[0, 0, 0], [1, 0, 0], [0, 1, 0], [1, 1, 0], [0, 0, 1], [1, 0, 1], [0, 1, 1], [1, 1, 1]] 8

mots_equ(4)
E(4)

[[0, 0, 0, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [1, 0, 1, 0], [0, 1, 1, 0], [1, 1, 1, 0], [0, 0, 0, 1], [1, 0, 0, 1], [0, 1, 0, 1], [1, 1, 0, 1], [1, 0, 1, 1], [0, 1, 1, 1], [1, 1, 1, 1]] 14

mots_equ(5)
E(5)

[[0, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [1, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0], [1, 1, 0, 1, 0], [1, 0, 1, 1, 0], [0, 1, 1, 1, 0], [1, 1, 1, 1, 0], [0, 0, 0, 0, 1], [1, 0, 0, 0, 1], [0, 1, 0, 0, 1], [0, 0, 1, 0, 1], [1, 0, 1, 0, 1], [0, 1, 1, 0, 1], [1, 1, 1, 0, 1], [0, 1, 0, 1, 1], [1, 1, 0, 1, 1], [1, 0, 1, 1, 1], [0, 1, 1, 1, 1], [1, 1, 1, 1, 1]] 24

mots_equ(6)
E(6)

[[0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [1, 0, 0, 1, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0], [1, 0, 1, 0, 1, 0], [0, 1, 1, 0, 1, 0], [0, 1, 0, 1, 1, 0], [1, 1, 0, 1, 1, 0], [1, 0, 1, 1, 1, 0], [0, 1, 1, 1, 1, 0], [1, 1, 1, 1, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 1], [0, 0, 1, 0, 0, 1], [1, 0, 1, 0, 0, 1], [1, 0, 0, 1, 0, 1], [0, 1, 0, 1, 0, 1], [1, 1, 0, 1, 0, 1], [1, 0, 1, 1, 0, 1], [0, 1, 1, 1, 0, 1], [1, 1, 1, 1, 0, 1], [1, 0, 1, 0, 1, 1], [0, 1, 1, 0, 1, 1], [1, 1, 1, 0, 1, 1], [1, 1, 0, 1, 1, 1], [1, 0, 1, 1, 1, 1], [0, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]] 36

limit?

File: /projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/calculus/calculus.py
Signature : limit(ex, dir=None, taylor=False, algorithm='maxima', **kwds)
Docstring :
Return the limit as the variable v approaches a from the given
direction.

   expr.limit(x = a)
   expr.limit(x = a, dir='+')

INPUT:

* "dir" - (default: None); dir may have the value 'plus' (or '+'
  or 'right') for a limit from above, 'minus' (or '-' or 'left')
  for a limit from below, or may be omitted (implying a two-sided
  limit is to be computed).

* "taylor" - (default: False); if True, use Taylor series, which
  allows more limits to be computed (but may also crash in some
  obscure cases due to bugs in Maxima).

* "**argv" - 1 named parameter

Note: The output may also use 'und' (undefined), 'ind'
  (indefinite but bounded), and 'infinity' (complex infinity).

EXAMPLES:

   sage: x = var('x')
   sage: f = (1+1/x)^x
   sage: f.limit(x = oo)
   e
   sage: f.limit(x = 5)
   7776/3125
   sage: f.limit(x = 1.2)
   2.06961575467...
   sage: f.limit(x = I, taylor=True)
   (-I + 1)^I
   sage: f(x=1.2)
   2.0696157546720...
   sage: f(x=I)
   (-I + 1)^I
   sage: CDF(f(x=I))
   2.0628722350809046 + 0.7450070621797239*I
   sage: CDF(f.limit(x = I))
   2.0628722350809046 + 0.7450070621797239*I

Notice that Maxima may ask for more information:

   sage: var('a')
   a
   sage: limit(x^a,x=0)
   Traceback (most recent call last):
   ...
   ValueError: Computation failed since Maxima requested additional
   constraints; using the 'assume' command before evaluation
   *may* help (example of legal syntax is 'assume(a>0)', see
   `assume?` for more details)
   Is a positive, negative or zero?

With this example, Maxima is looking for a LOT of information:

   sage: assume(a>0)
   sage: limit(x^a,x=0)
   Traceback (most recent call last):
   ...
   ValueError: Computation failed since Maxima requested additional
   constraints; using the 'assume' command before evaluation *may* help
   (example of legal syntax is 'assume(a>0)', see `assume?` for
    more details)
   Is a an integer?
   sage: assume(a,'integer')
   sage: limit(x^a,x=0)
   Traceback (most recent call last):
   ...
   ValueError: Computation failed since Maxima requested additional
   constraints; using the 'assume' command before evaluation *may* help
   (example of legal syntax is 'assume(a>0)', see `assume?` for
    more details)
   Is a an even number?
   sage: assume(a,'even')
   sage: limit(x^a,x=0)
   0
   sage: forget()

More examples:

   sage: limit(x*log(x), x = 0, dir='+')
   0
   sage: lim((x+1)^(1/x), x = 0)
   e
   sage: lim(e^x/x, x = oo)
   +Infinity
   sage: lim(e^x/x, x = -oo)
   0
   sage: lim(-e^x/x, x = oo)
   -Infinity
   sage: lim((cos(x))/(x^2), x = 0)
   +Infinity
   sage: lim(sqrt(x^2+1) - x, x = oo)
   0
   sage: lim(x^2/(sec(x)-1), x=0)
   2
   sage: lim(cos(x)/(cos(x)-1), x=0)
   -Infinity
   sage: lim(x*sin(1/x), x=0)
   0
   sage: limit(e^(-1/x), x=0, dir='right')
   0
   sage: limit(e^(-1/x), x=0, dir='left')
   +Infinity

   sage: f = log(log(x))/log(x)
   sage: forget(); assume(x<-2); lim(f, x=0, taylor=True)
   0
   sage: forget()

Here ind means "indefinite but bounded":

   sage: lim(sin(1/x), x = 0)
   ind

a=-5
b=4
n=var('n')
f=(a^n+b^n)^(1/n)
f.limit(n=Infinity)

limit((4^n + (-5)^n)^(1/n), n, +Infinity)

((-3)^11)^(1/11)

3*(-1)^(1/11)

4^3>3^4

False

1/(2*3)>(1/(2*4)+(6/(2^2*4^4)))

True

3*((1/(2*3^2)) )>4*((1/(2*4^2)) +(1/(2*4^2))^3)+(6/(2*4^2))^2

True

binomial(5,4)

5

6*(1/(2*4^2))^2  > 10*((1/(2*5^2))^2+(1/(2*5^2))^3)

True

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