def test_evalf_integer_parts():
a = floor(log(8)/log(2) - exp(-1000), evaluate=False)
b = floor(log(8)/log(2), evaluate=False)
assert a.evalf() == 3
assert b.evalf() == 3
# equals, as a fallback, can still fail but it might succeed as here
assert ceiling(10*(sin(1)**2 + cos(1)**2)) == 10
assert int(floor(factorial(50)/E, evaluate=False).evalf(70)) == \
long(11188719610782480504630258070757734324011354208865721592720336800)
assert int(ceiling(factorial(50)/E, evaluate=False).evalf(70)) == \
long(11188719610782480504630258070757734324011354208865721592720336801)
assert int(floor((GoldenRatio**999 / sqrt(5) + Rational(1, 2)))
.evalf(1000)) == fibonacci(999)
assert int(floor((GoldenRatio**1000 / sqrt(5) + Rational(1, 2)))
.evalf(1000)) == fibonacci(1000)
assert ceiling(x).evalf(subs={x: 3}) == 3
assert ceiling(x).evalf(subs={x: 3*I}) == 3*I
assert ceiling(x).evalf(subs={x: 2 + 3*I}) == 2 + 3*I
assert ceiling(x).evalf(subs={x: 3.}) == 3
assert ceiling(x).evalf(subs={x: 3.*I}) == 3*I
assert ceiling(x).evalf(subs={x: 2. + 3*I}) == 2 + 3*I
assert float((floor(1.5, evaluate=False)+1/9).evalf()) == 1 + 1/9
assert float((floor(0.5, evaluate=False)+20).evalf()) == 20
def test_evalf_integer_parts():
a = floor(log(8) / log(2) - exp(-1000), evaluate=False)
b = floor(log(8) / log(2), evaluate=False)
assert a.evalf() == 3
assert b.evalf() == 3
# equals, as a fallback, can still fail but it might succeed as here
assert ceiling(10 * (sin(1)**2 + cos(1)**2)) == 10
assert int(floor(factorial(50)/E, evaluate=False).evalf(70)) == \
long(11188719610782480504630258070757734324011354208865721592720336800)
assert int(ceiling(factorial(50)/E, evaluate=False).evalf(70)) == \
long(11188719610782480504630258070757734324011354208865721592720336801)
assert int(
floor((GoldenRatio**999 / sqrt(5) +
Rational(1, 2))).evalf(1000)) == fibonacci(999)
assert int(
floor((GoldenRatio**1000 / sqrt(5) +
Rational(1, 2))).evalf(1000)) == fibonacci(1000)
assert ceiling(x).evalf(subs={x: 3}) == 3
assert ceiling(x).evalf(subs={x: 3 * I}) == 3 * I
assert ceiling(x).evalf(subs={x: 2 + 3 * I}) == 2 + 3 * I
assert ceiling(x).evalf(subs={x: 3.}) == 3
assert ceiling(x).evalf(subs={x: 3. * I}) == 3 * I
assert ceiling(x).evalf(subs={x: 2. + 3 * I}) == 2 + 3 * I
assert float((floor(1.5, evaluate=False) + 1 / 9).evalf()) == 1 + 1 / 9
assert float((floor(0.5, evaluate=False) + 20).evalf()) == 20
def _compute_rho(self, k, num_iterations):
# The l variable counts backwards whereas k counts forwards.
l = num_iterations - (k-1)
# Compute rho using Fibonacci numbers
result = 1 - sy.fibonacci(l+1) / sy.fibonacci(l+2)
return float(result)
def test_linrec():
assert linrec(coeffs=[1, 1], init=[1, 1], n=20) == 10946
assert linrec(coeffs=[1, 2, 3, 4, 5], init=[1, 1, 0, 2], n=10) == 1040
assert linrec(coeffs=[0, 0, 11, 13], init=[23, 27], n=25) == 59628567384
assert linrec(coeffs=[0, 0, 1, 1, 2], init=[1, 5, 3], n=15) == 165
assert linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4], n=70) == \
56889923441670659718376223533331214868804815612050381493741233489928913241
assert linrec(coeffs=[0]*55 + [1, 1, 2, 3], init=[0]*50 + [1, 2, 3], n=4000) == \
702633573874937994980598979769135096432444135301118916539
assert linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4], n=10**4)
assert linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4], n=10**5)
assert all(
linrec(coeffs=[1, 1], init=[0, 1], n=n) == fibonacci(n)
for n in range(95, 115))
assert all(
linrec(coeffs=[1, 1], init=[1, 1], n=n) == fibonacci(n + 1)
for n in range(595, 615))
a = [S(1) / 2, S(3) / 4, S(5) / 6, 7, S(8) / 9, S(3) / 5]
b = [1, 2, 8, S(5) / 7, S(3) / 7, S(2) / 9, 6]
x, y, z = symbols('x y z')
assert linrec(coeffs=a[:5], init=b[:4], n=80) == \
Rational(1726244235456268979436592226626304376013002142588105090705187189,
1960143456748895967474334873705475211264)
assert linrec(coeffs=a[:4], init=b[:4], n=50) == \
Rational(368949940033050147080268092104304441, 504857282956046106624)
assert linrec(coeffs=a[3:], init=b[:3], n=35) == \
Rational(97409272177295731943657945116791049305244422833125109,
814315512679031689453125)
assert linrec(coeffs=[0]*60 + [S(2)/3, S(4)/5], init=b, n=3000) == \
26777668739896791448594650497024/S(48084516708184142230517578125)
raises(TypeError,
lambda: linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4, 5], n=1))
raises(TypeError, lambda: linrec(coeffs=a[:4], init=b[:5], n=10000))
raises(ValueError, lambda: linrec(coeffs=a[:4], init=b[:4], n=-10000))
raises(TypeError, lambda: linrec(x, b, n=10000))
raises(TypeError, lambda: linrec(a, y, n=10000))
assert linrec(coeffs=[x, y, z], init=[1, 1, 1], n=4) == \
x**2 + x*y + x*z + y + z
assert linrec(coeffs=[1, 2, 1], init=[x, y, z], n=20) == \
269542*x + 664575*y + 578949*z
assert linrec(coeffs=[0, 3, 1, 2], init=[x, y], n=30) == \
58516436*x + 56372788*y
assert linrec(coeffs=[0]*50 + [1, 2, 3], init=[x, y, z], n=1000) == \
11477135884896*x + 25999077948732*y + 41975630244216*z
assert linrec(coeffs=[], init=[1, 1], n=20) == 0
def test_linrec():
assert linrec(coeffs=[1, 1], init=[1, 1], n=20) == 10946
assert linrec(coeffs=[1, 2, 3, 4, 5], init=[1, 1, 0, 2], n=10) == 1040
assert linrec(coeffs=[0, 0, 11, 13], init=[23, 27], n=25) == 59628567384
assert linrec(coeffs=[0, 0, 1, 1, 2], init=[1, 5, 3], n=15) == 165
assert linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4], n=70) == \
56889923441670659718376223533331214868804815612050381493741233489928913241
assert linrec(coeffs=[0]*55 + [1, 1, 2, 3], init=[0]*50 + [1, 2, 3], n=4000) == \
702633573874937994980598979769135096432444135301118916539
assert linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4], n=10**4)
assert linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4], n=10**5)
assert all(linrec(coeffs=[1, 1], init=[0, 1], n=n) == fibonacci(n)
for n in range(95, 115))
assert all(linrec(coeffs=[1, 1], init=[1, 1], n=n) == fibonacci(n + 1)
for n in range(595, 615))
a = [S(1)/2, S(3)/4, S(5)/6, 7, S(8)/9, S(3)/5]
b = [1, 2, 8, S(5)/7, S(3)/7, S(2)/9, 6]
x, y, z = symbols('x y z')
assert linrec(coeffs=a[:5], init=b[:4], n=80) == \
Rational(1726244235456268979436592226626304376013002142588105090705187189,
1960143456748895967474334873705475211264)
assert linrec(coeffs=a[:4], init=b[:4], n=50) == \
Rational(368949940033050147080268092104304441, 504857282956046106624)
assert linrec(coeffs=a[3:], init=b[:3], n=35) == \
Rational(97409272177295731943657945116791049305244422833125109,
814315512679031689453125)
assert linrec(coeffs=[0]*60 + [S(2)/3, S(4)/5], init=b, n=3000) == \
26777668739896791448594650497024/S(48084516708184142230517578125)
raises(TypeError, lambda: linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4, 5], n=1))
raises(TypeError, lambda: linrec(coeffs=a[:4], init=b[:5], n=10000))
raises(ValueError, lambda: linrec(coeffs=a[:4], init=b[:4], n=-10000))
raises(TypeError, lambda: linrec(x, b, n=10000))
raises(TypeError, lambda: linrec(a, y, n=10000))
assert linrec(coeffs=[x, y, z], init=[1, 1, 1], n=4) == \
x**2 + x*y + x*z + y + z
assert linrec(coeffs=[1, 2, 1], init=[x, y, z], n=20) == \
269542*x + 664575*y + 578949*z
assert linrec(coeffs=[0, 3, 1, 2], init=[x, y], n=30) == \
58516436*x + 56372788*y
assert linrec(coeffs=[0]*50 + [1, 2, 3], init=[x, y, z], n=1000) == \
11477135884896*x + 25999077948732*y + 41975630244216*z
def test_evalf_integer_parts():
a = floor(log(8)/log(2) - exp(-1000), evaluate=False)
b = floor(log(8)/log(2), evaluate=False)
raises(PrecisionExhausted, "a.evalf()")
assert a.evalf(chop=True) == 3
assert a.evalf(maxprec=500) == 2
raises(PrecisionExhausted, "b.evalf()")
raises(PrecisionExhausted, "b.evalf(maxprec=500)")
assert b.evalf(chop=True) == 3
assert int(floor(factorial(50)/E,evaluate=False).evalf()) == \
11188719610782480504630258070757734324011354208865721592720336800L
assert int(ceiling(factorial(50)/E,evaluate=False).evalf()) == \
11188719610782480504630258070757734324011354208865721592720336801L
assert int(floor((GoldenRatio**999 / sqrt(5) + Rational(1,2))).evalf(1000)) == fibonacci(999)
assert int(floor((GoldenRatio**1000 / sqrt(5) + Rational(1,2))).evalf(1000)) == fibonacci(1000)
def test_evalf_integer_parts():
a = floor(log(8)/log(2) - exp(-1000), evaluate=False)
b = floor(log(8)/log(2), evaluate=False)
raises(PrecisionExhausted, "a.evalf()")
assert a.evalf(chop=True) == 3
assert a.evalf(maxn=500) == 2
raises(PrecisionExhausted, "b.evalf()")
raises(PrecisionExhausted, "b.evalf(maxn=500)")
assert b.evalf(chop=True) == 3
assert int(floor(factorial(50)/E,evaluate=False).evalf()) == \
11188719610782480504630258070757734324011354208865721592720336800L
assert int(ceiling(factorial(50)/E,evaluate=False).evalf()) == \
11188719610782480504630258070757734324011354208865721592720336801L
assert int(floor((GoldenRatio**999 / sqrt(5) + Rational(1,2))).evalf(1000)) == fibonacci(999)
assert int(floor((GoldenRatio**1000 / sqrt(5) + Rational(1,2))).evalf(1000)) == fibonacci(1000)
def test_find_simple_recurrence_vector():
assert find_simple_recurrence_vector([fibonacci(k)
for k in range(12)]) == [
1,
-1,
-1,
]
def test_evalf_integer_parts():
a = floor(log(8)/log(2) - exp(-1000), evaluate=False)
b = floor(log(8)/log(2), evaluate=False)
raises(PrecisionExhausted, "a.evalf()")
assert a.evalf(chop=True) == 3
assert a.evalf(maxn=500) == 2
assert b.evalf() == 3
# equals, as a fallback, can still fail but it might succeed as here
assert ceiling(10*(sin(1)**2 + cos(1)**2)) == 10
assert int(floor(factorial(50)/E,evaluate=False).evalf()) == \
11188719610782480504630258070757734324011354208865721592720336800L
assert int(ceiling(factorial(50)/E,evaluate=False).evalf()) == \
11188719610782480504630258070757734324011354208865721592720336801L
assert int(floor((GoldenRatio**999 / sqrt(5) + Rational(1,2))).evalf(1000)) == fibonacci(999)
assert int(floor((GoldenRatio**1000 / sqrt(5) + Rational(1,2))).evalf(1000)) == fibonacci(1000)
def test_tenth_fibonacci_tuple():
"""Ensure that the request for the tenth Fibonacci number returns the correct tuple of numbers."""
number = 10
result = fibonacci.fibonacci_tuple(number)
assert len(result) == number
assert result == (1, 1, 2, 3, 5, 8, 13, 21, 34, 55)
assert sympy.fibonacci(10) == result[-1]
def test_second_fibonacci_tuple():
"""Ensure that the request for the second Fibonacci number returns same number twice in a tuple."""
number = 2
result = fibonacci.fibonacci_tuple(number)
assert len(result) == number
assert result == (1, 1)
assert sympy.fibonacci(2) == result[-1]
def test_first_fibonacci_singleton_tuple():
"""Ensure that the request for first Fibonacci number returns same number in a tuple."""
number = 1
result = fibonacci.fibonacci_tuple(number)
assert len(result) == number
assert result == (1, )
assert sympy.fibonacci(1) == result[-1]
def test_evalf_integer_parts():
a = floor(log(8)/log(2) - exp(-1000), evaluate=False)
b = floor(log(8)/log(2), evaluate=False)
raises(PrecisionExhausted, lambda: a.evalf())
assert a.evalf(chop=True) == 3
assert a.evalf(maxn=500) == 2
assert b.evalf() == 3
# equals, as a fallback, can still fail but it might succeed as here
assert ceiling(10*(sin(1)**2 + cos(1)**2)) == 10
assert int(floor(factorial(50)/E, evaluate=False).evalf(70)) == \
long(11188719610782480504630258070757734324011354208865721592720336800)
assert int(ceiling(factorial(50)/E, evaluate=False).evalf(70)) == \
long(11188719610782480504630258070757734324011354208865721592720336801)
assert int(floor((GoldenRatio**999 / sqrt(5) + Rational(1, 2)))
.evalf(1000)) == fibonacci(999)
assert int(floor((GoldenRatio**1000 / sqrt(5) + Rational(1, 2)))
.evalf(1000)) == fibonacci(1000)
def test_evalf_near_integers():
# Binet's formula
f = lambda n: ((1 + sqrt(5))**n)/(2**n * sqrt(5))
assert NS(f(5000) - fibonacci(5000), 10, maxn=1500) == '5.156009964e-1046'
# Some near-integer identities from
# http://mathworld.wolfram.com/AlmostInteger.html
assert NS('sin(2017*2**(1/5))', 15) == '-1.00000000000000'
assert NS('sin(2017*2**(1/5))', 20) == '-0.99999999999999997857'
assert NS('1+sin(2017*2**(1/5))', 15) == '2.14322287389390e-17'
assert NS('45 - 613*E/37 + 35/991', 15) == '6.03764498766326e-11'
def test_evalf_near_integers():
# Binet's formula
f = lambda n: ((1 + sqrt(5)) ** n) / (2 ** n * sqrt(5))
assert NS(f(5000) - fibonacci(5000), 10, maxprec=1500) == "5.156009964e-1046"
# Some near-integer identities from
# http://mathworld.wolfram.com/AlmostInteger.html
assert NS("sin(2017*2**(1/5))", 15) == "-1.00000000000000"
assert NS("sin(2017*2**(1/5))", 20) == "-0.99999999999999997857"
assert NS("1+sin(2017*2**(1/5))", 15) == "2.14322287389390e-17"
assert NS("45 - 613*E/37 + 35/991", 15) == "6.03764498766326e-11"
def test_guess_generating_function():
x = Symbol('x')
assert guess_generating_function([fibonacci(k)
for k in range(5, 15)]) == ((3*x + 5)/(-x**2 - x + 1))
assert guess_generating_function(
[1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122, 38919]) == (
(1/(x**4 + 2*x**2 - 4*x + 1))**(sympify("1/2")))
assert guess_generating_function(sympify(
"[3/2, 11/2, 0, -121/2, -363/2, 121, 4719/2, 11495/2, -8712, -178717/2]")
) == (x + sympify("3/2"))/(11*x**2 - 3*x + 1)
def test_guess_generating_function():
x = Symbol('x')
assert guess_generating_function([fibonacci(k) for k in range(5, 15)
]) == ((3 * x + 5) / (-x**2 - x + 1))
assert guess_generating_function(
[1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122,
38919]) == ((1 / (x**4 + 2 * x**2 - 4 * x + 1))**(sympify("1/2")))
assert guess_generating_function(
sympify(
"[3/2, 11/2, 0, -121/2, -363/2, 121, 4719/2, 11495/2, -8712, -178717/2]"
)) == (x + sympify("3/2")) / (11 * x**2 - 3 * x + 1)
def test_guess_generating_function():
x = Symbol('x')
assert guess_generating_function([fibonacci(k)
for k in range(5, 15)])['ogf'] == ((3*x + 5)/(-x**2 - x + 1))
assert guess_generating_function(
[1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122, 38919])['ogf'] == (
(1/(x**4 + 2*x**2 - 4*x + 1))**Rational(1, 2))
assert guess_generating_function(sympify(
"[3/2, 11/2, 0, -121/2, -363/2, 121, 4719/2, 11495/2, -8712, -178717/2]")
)['ogf'] == (x + Rational(3, 2))/(11*x**2 - 3*x + 1)
assert guess_generating_function([factorial(k) for k in range(12)],
types=['egf'])['egf'] == 1/(-x + 1)
assert guess_generating_function([k+1 for k in range(12)],
types=['egf']) == {'egf': (x + 1)*exp(x), 'lgdegf': (x + 2)/(x + 1)}
def test_find_simple_recurrence():
a = Function('a')
n = Symbol('n')
assert find_simple_recurrence([fibonacci(k) for k in range(12)]) == (
-a(n) - a(n + 1) + a(n + 2))
f = Function('a')
i = Symbol('n')
a = [1, 1, 1]
for k in range(15): a.append(5*a[-1]-3*a[-2]+8*a[-3])
assert find_simple_recurrence(a, A=f, N=i) == (
-8*f(i) + 3*f(i + 1) - 5*f(i + 2) + f(i + 3))
assert find_simple_recurrence([0, 2, 15, 74, 12, 3, 0,
1, 2, 85, 4, 5, 63]) == 0
def test_find_simple_recurrence():
a = Function('a')
n = Symbol('n')
assert find_simple_recurrence([fibonacci(k) for k in range(12)
]) == (-a(n) - a(n + 1) + a(n + 2))
f = Function('a')
i = Symbol('n')
a = [1, 1, 1]
for k in range(15):
a.append(5 * a[-1] - 3 * a[-2] + 8 * a[-3])
assert find_simple_recurrence(a, A=f, N=i) == (-8 * f(i) + 3 * f(i + 1) -
5 * f(i + 2) + f(i + 3))
assert find_simple_recurrence([0, 2, 15, 74, 12, 3, 0, 1, 2, 85, 4, 5,
63]) == 0
def test_evalf_integer_parts():
a = floor(log(8) / log(2) - exp(-1000), evaluate=False)
b = floor(log(8) / log(2), evaluate=False)
assert a.evalf() == 3
assert b.evalf() == 3
# equals, as a fallback, can still fail but it might succeed as here
assert ceiling(10 * (sin(1)**2 + cos(1)**2)) == 10
assert int(floor(factorial(50)/E, evaluate=False).evalf(70)) == \
int(11188719610782480504630258070757734324011354208865721592720336800)
assert int(ceiling(factorial(50)/E, evaluate=False).evalf(70)) == \
int(11188719610782480504630258070757734324011354208865721592720336801)
assert int(floor(GoldenRatio**999 / sqrt(5) +
S.Half).evalf(1000)) == fibonacci(999)
assert int(floor(GoldenRatio**1000 / sqrt(5) +
S.Half).evalf(1000)) == fibonacci(1000)
assert ceiling(x).evalf(subs={x: 3}) == 3
assert ceiling(x).evalf(subs={x: 3 * I}) == 3.0 * I
assert ceiling(x).evalf(subs={x: 2 + 3 * I}) == 2.0 + 3.0 * I
assert ceiling(x).evalf(subs={x: 3.}) == 3
assert ceiling(x).evalf(subs={x: 3. * I}) == 3.0 * I
assert ceiling(x).evalf(subs={x: 2. + 3 * I}) == 2.0 + 3.0 * I
assert float((floor(1.5, evaluate=False) + 1 / 9).evalf()) == 1 + 1 / 9
assert float((floor(0.5, evaluate=False) + 20).evalf()) == 20
# issue 19991
n = 1169809367327212570704813632106852886389036911
r = 744723773141314414542111064094745678855643068
assert floor(n / (pi / 2)) == r
assert floor(80782 * sqrt(2)) == 114242
# issue 20076
assert 260515 - floor(260515 / pi + 1 / 2) * pi == atan(tan(260515))
def test_fibonacci():
assert [fibonacci(n) for n in range(-3, 5)] == [2, -1, 1, 0, 1, 1, 2, 3]
assert fibonacci(100) == 354224848179261915075
assert [lucas(n) for n in range(-3, 5)] == [-4, 3, -1, 2, 1, 3, 4, 7]
assert lucas(100) == 792070839848372253127
assert fibonacci(1, x) == 1
assert fibonacci(2, x) == x
assert fibonacci(3, x) == x**2 + 1
assert fibonacci(4, x) == x**3 + 2 * x
def test_fibonacci():
assert [fibonacci(n) for n in range(-3, 5)] == [2, -1, 1, 0, 1, 1, 2, 3]
assert fibonacci(100) == 354224848179261915075
assert [lucas(n) for n in range(-3, 5)] == [-4, 3, -1, 2, 1, 3, 4, 7]
assert lucas(100) == 792070839848372253127
assert fibonacci(1, x) == 1
assert fibonacci(2, x) == x
assert fibonacci(3, x) == x**2 + 1
assert fibonacci(4, x) == x**3 + 2*x
def test_approximants():
x, t = symbols("x,t")
g = [lucas(k) for k in range(16)]
assert [e for e in approximants(g)] == (
[2, -4/(x - 2), (5*x - 2)/(3*x - 1), (x - 2)/(x**2 + x - 1)] )
g = [lucas(k)+fibonacci(k+2) for k in range(16)]
assert [e for e in approximants(g)] == (
[3, -3/(x - 1), (3*x - 3)/(2*x - 1), -3/(x**2 + x - 1)] )
g = [lucas(k)**2 for k in range(16)]
assert [e for e in approximants(g)] == (
[4, -16/(x - 4), (35*x - 4)/(9*x - 1), (37*x - 28)/(13*x**2 + 11*x - 7),
(50*x**2 + 63*x - 52)/(37*x**2 + 19*x - 13),
(-x**2 - 7*x + 4)/(x**3 - 2*x**2 - 2*x + 1)] )
p = [sum(binomial(k,i)*x**i for i in range(k+1)) for k in range(16)]
y = approximants(p, t, simplify=True)
assert next(y) == 1
assert next(y) == -1/(t*(x + 1) - 1)
def test_approximants():
x, t = symbols("x,t")
g = [lucas(k) for k in range(16)]
assert [e for e in approximants(g)] == (
[2, -4/(x - 2), (5*x - 2)/(3*x - 1), (x - 2)/(x**2 + x - 1)] )
g = [lucas(k)+fibonacci(k+2) for k in range(16)]
assert [e for e in approximants(g)] == (
[3, -3/(x - 1), (3*x - 3)/(2*x - 1), -3/(x**2 + x - 1)] )
g = [lucas(k)**2 for k in range(16)]
assert [e for e in approximants(g)] == (
[4, -16/(x - 4), (35*x - 4)/(9*x - 1), (37*x - 28)/(13*x**2 + 11*x - 7),
(50*x**2 + 63*x - 52)/(37*x**2 + 19*x - 13),
(-x**2 - 7*x + 4)/(x**3 - 2*x**2 - 2*x + 1)] )
p = [sum(binomial(k,i)*x**i for i in range(k+1)) for k in range(16)]
y = approximants(p, t, simplify=True)
assert next(y) == 1
assert next(y) == -1/(t*(x + 1) - 1)
def estimate_iterations(self):
"""
Estimates the number of iterations required to find the
value of x within a range of the specified uncertainty.
"""
N = sy.symbols('N')
initial_range = self._x_end - self._x_start
final_range = self._uncertainty
ineq = N >= (1 + 2*self._epsilon) / (final_range / initial_range)
interval = sy.solveset(ineq, N, sy.S.Reals)
interval_start = int(np.ceil(float(interval.args[0])))
for i in range(2, 10):
if interval_start <= sy.fibonacci(i):
return i-2
raise ValueError('Could not find the correct Fibonacci number.')
def inverse_secant(p, a, n, x0, x1):
l = [fibonacci(i) for i in range(3, 20)]
j = 0
i = l[j]
while i < n:
s = (a * (x1 * x0)) % (p**i)
x = x1 + x0
x0 = x1
x1 = x
x1 -= s
j += 1
i = l[j]
s = (a * (x1 * x0)) % (p**n)
x = x1 + x0
x0 = x1
x1 = x
x1 -= s
return (x1 % (p**n))
def test_guess_generating_function():
x = Symbol('x')
assert guess_generating_function(
[fibonacci(k)
for k in range(5, 15)])['ogf'] == ((3 * x + 5) / (-x**2 - x + 1))
assert guess_generating_function(
[1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122,
38919])['ogf'] == ((1 / (x**4 + 2 * x**2 - 4 * x + 1))**S.Half)
assert guess_generating_function(
sympify(
"[3/2, 11/2, 0, -121/2, -363/2, 121, 4719/2, 11495/2, -8712, -178717/2]"
))['ogf'] == (x + Rational(3, 2)) / (11 * x**2 - 3 * x + 1)
assert guess_generating_function([factorial(k) for k in range(12)],
types=['egf'])['egf'] == 1 / (-x + 1)
assert guess_generating_function([k + 1 for k in range(12)],
types=['egf']) == {
'egf': (x + 1) * exp(x),
'lgdegf': (x + 2) / (x + 1)
}
def main():
fibo_var = 39
arr_fibo = []
for i in range(1,fibo_var):
arr_fibo.append(sp.fibonacci(i))
#base will be a number in arr_fibo
#ht will be a number which is +- 1 of base
#(L**2) = (ht**2) + ((base*0.5)**2)
#print arr_fibo
#Calculate the difference betweeen the squares of adjacent numbers
arr_base = []
for i in range(1,len(arr_fibo)):
temp = ((arr_fibo[i]**2) - (arr_fibo[i-1]**2))
arr_base.append(temp)
#Remove zeros from the array
for i in range(0,len(arr_base)-1):
if(arr_base[i] == 0):
arr_base.remove(0)
summa = 0
for i in range(0,len(arr_base)):
ht = arr_base[i] + 1
base = arr_base[i]
var = math.sqrt(((base*0.5)**2) + (ht**2))
if(var.is_integer()):
#print int(var)
summa += int(var)
####################
ht = arr_base[i] - 1
base = arr_base[i]
var = math.sqrt(((base*0.5)**2) + (ht**2))
if(var.is_integer()):
#print int(var)
summa += int(var)
print summa
def test_find_linear_recurrence():
assert sequence((0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55), \
(n, 0, 10)).find_linear_recurrence(11) == [1, 1]
assert sequence((1, 2, 4, 7, 28, 128, 582, 2745, 13021, 61699, 292521, \
1387138), (n, 0, 11)).find_linear_recurrence(12) == [5, -2, 6, -11]
assert sequence(x*n**3+y*n, (n, 0, oo)).find_linear_recurrence(10) \
== [4, -6, 4, -1]
assert sequence(x**n, (n, 0, 20)).find_linear_recurrence(21) == [x]
assert sequence((1, 2, 3)).find_linear_recurrence(10, 5) == [0, 0, 1]
assert sequence(((1 + sqrt(5))/2)**n + \
(-(1 + sqrt(5))/2)**(-n)).find_linear_recurrence(10) == [1, 1]
assert sequence(x*((1 + sqrt(5))/2)**n + y*(-(1 + sqrt(5))/2)**(-n), \
(n,0,oo)).find_linear_recurrence(10) == [1, 1]
assert sequence((1, 2, 3, 4, 6), (n, 0, 4)).find_linear_recurrence(5) == []
assert sequence((2,3,4,5,6,79),(n, 0, 5)).find_linear_recurrence(6,gfvar=x) \
== ([], None)
assert sequence((2,3,4,5,8,30),(n, 0, 5)).find_linear_recurrence(6,gfvar=x) \
== ([19/2, -20, 27/2], (-31*x**2 + 32*x - 4)/(27*x**3 - 40*x**2 + 19*x -2))
assert sequence(fibonacci(n)).find_linear_recurrence(30,gfvar=x) \
== ([1, 1], -x/(x**2 + x - 1))
def test_find_linear_recurrence():
assert sequence((0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55), \
(n, 0, 10)).find_linear_recurrence(11) == [1, 1]
assert sequence((1, 2, 4, 7, 28, 128, 582, 2745, 13021, 61699, 292521, \
1387138), (n, 0, 11)).find_linear_recurrence(12) == [5, -2, 6, -11]
assert sequence(x*n**3+y*n, (n, 0, oo)).find_linear_recurrence(10) \
== [4, -6, 4, -1]
assert sequence(x**n, (n,0,20)).find_linear_recurrence(21) == [x]
assert sequence((1,2,3)).find_linear_recurrence(10, 5) == [0, 0, 1]
assert sequence(((1 + sqrt(5))/2)**n + \
(-(1 + sqrt(5))/2)**(-n)).find_linear_recurrence(10) == [1, 1]
assert sequence(x*((1 + sqrt(5))/2)**n + y*(-(1 + sqrt(5))/2)**(-n), \
(n,0,oo)).find_linear_recurrence(10) == [1, 1]
assert sequence((1,2,3,4,6),(n, 0, 4)).find_linear_recurrence(5) == []
assert sequence((2,3,4,5,6,79),(n, 0, 5)).find_linear_recurrence(6,gfvar=x) \
== ([], None)
assert sequence((2,3,4,5,8,30),(n, 0, 5)).find_linear_recurrence(6,gfvar=x) \
== ([19/2, -20, 27/2], (-31*x**2 + 32*x - 4)/(27*x**3 - 40*x**2 + 19*x -2))
assert sequence(fibonacci(n)).find_linear_recurrence(30,gfvar=x) \
== ([1, 1], -x/(x**2 + x - 1))
def test_guess_generating_function_rational():
x = Symbol('x')
assert guess_generating_function_rational([fibonacci(k)
for k in range(5, 15)]) == ((3*x + 5)/(-x**2 - x + 1))
def apply(self, n, evaluation):
'Fibonacci[n_Integer]'
return Integer(sympy.fibonacci(n.get_int_value()))
def test_guess_generating_function_rational():
x = Symbol('x')
assert guess_generating_function_rational([
fibonacci(k) for k in range(5, 15)
]) == ((3 * x + 5) / (-x**2 - x + 1))
def apply(self, n, evaluation):
'Fibonacci[n_Integer]'
return Integer(sympy.fibonacci(n.get_int_value()))
def fibonacci_numbers(start=0, limit=100, symbolic_keys=True):
f = IndexedBase('f')
return {(f[i] if symbolic_keys else i):fibonacci(i) for i in range(start, limit)}
def apply(self, n, evaluation):
'Fibonacci[n_Integer]'
return Integer(sympy.fibonacci(n.to_sympy()))
def apply(self, n, evaluation):
'Fibonacci[n_Integer]'
return Integer(sympy.fibonacci(n.to_sympy()))
""" Created Nov 15, 2012 Author: Spencer Lyon Project Euler Problem 25 """ import sympy as sym import numpy as np fibs = [str(sym.fibonacci(n)) for n in xrange(5000)] lens = np.array([len(fibs[i]) for i in xrange(5000)]) ans = np.where(lens == 1000)[0][0] print 'The answer is: ', ans
from sympy import fibonacci
maximum = 4000000
total = 0
n = 0
while (fibonacci(n) < maximum):
if (fibonacci(n).is_even):
total += fibonacci(n)
n += 1
print(total)
def test_issue_10382():
n = Symbol('n', integer=True)
assert limit_seq(fibonacci(n+1)/fibonacci(n), n) == S.GoldenRatio
def test_issue_10382():
n = Symbol('n', integer=True)
assert limit_seq(fibonacci(n + 1) / fibonacci(n), n) == S.GoldenRatio
def test_find_simple_recurrence_vector():
assert find_simple_recurrence_vector(
[fibonacci(k) for k in range(12)]) == [1, -1, -1]
# SOLUTION: 😁
def a(n):
return (sp.fibonacci(n + 1) + sp.lucas(n)) / (sp.fibonacci(n + 2) +
sp.lucas(n + 1))
cnt = 0
sum = 0
n = 1
while cnt < 30:
cnt += 1
print(cnt, (3 - 2 * a(2 * n - 1)) / (1 - a(2 * n - 1) -
(a(2 * n - 1) * a(2 * n - 1))) - 3)
cnt += 1
print(
cnt, 3 * sp.fibonacci(2 * n + 1)**2 -
2 * sp.fibonacci(2 * n) * sp.fibonacci(2 * n + 1) - 3)
sum += (3 - 2 * a(2 * n - 1)) / (1 - a(2 * n - 1) - (a(2 * n - 1) * a(
2 * n - 1))) - 3 + 3 * sp.fibonacci(2 * n + 1)**2 - 2 * sp.fibonacci(
2 * n) * sp.fibonacci(2 * n + 1) - 3
n += 1
print("Ans: ", sum)
"""
1 2
2 5
3 21
4 42
5 152
6 296
7 1050
def test_fiftieth_fibonacci_tuple():
"""Ensure that the request for the 50th Fibonacci number is correct according to sympy function."""
number = 50
result = fibonacci.fibonacci_tuple(number)
assert len(result) == number
assert sympy.fibonacci(50) == result[-1]
def a(n):
return (sp.fibonacci(n + 1) + sp.lucas(n)) / (sp.fibonacci(n + 2) +
sp.lucas(n + 1))
from treys import Card, Evaluator
import extended as xt
import roman
import networkx as nx
evaluator = Evaluator()
getcontext().prec, getcontext().rounding = 2000, ROUND_DOWN
#%% Problem 1
#this returns the sum of the list from 0 to 999, divisible by either 3 or 5
np.sum([i for i in range(1000) if i%3==0 or i%5==0])
#%% Problem 2
#the inner brackets do the fibonacci sequence, which is filtered by the outer
#brackets. We then find the sum of the filtered list
int(np.sum([j for j in [sp.fibonacci(i) for i in range(1000)] if j<4*10**6 and j%2==0]))
#%% Problem 3
#simple sympy number theory
max(sp.primefactors(600851475143))
#%% Problem 4
#the inner bracket generates the products and the outer checks if the string of
#that number is equal to its reverse
max([k for k in [i*j for i in range(1,1000) for j in range(1,1000)] \
if str(k)==str(k)[::-1]])
#%% Problem 5
#numpy has this built in
np.lcm.reduce(range(1,20))
from sympy import fibonacci
maximum = 1000
index = 0
while (len(str(fibonacci(index))) != maximum):
index += 1
print(index)
