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from sympy.concrete.guess import (
find_simple_recurrence_vector,
find_simple_recurrence,
rationalize,
guess_generating_function_rational,
guess_generating_function,
guess
)
from sympy.concrete.products import Product
from sympy.core.function import Function
from sympy.core.numbers import Rational
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial)
from sympy.functions.combinatorial.numbers import fibonacci
from sympy.functions.elementary.exponential import exp
def test_find_simple_recurrence_vector():
assert find_simple_recurrence_vector(
[fibonacci(k) for k in range(12)]) == [1, -1, -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_rationalize():
from mpmath import cos, pi, mpf
assert rationalize(cos(pi/3)) == S.Half
assert rationalize(mpf("0.333333333333333")) == Rational(1, 3)
assert rationalize(mpf("-0.333333333333333")) == Rational(-1, 3)
assert rationalize(pi, maxcoeff = 250) == Rational(355, 113)
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 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 test_guess():
i0, i1 = symbols('i0 i1')
assert guess([1, 2, 6, 24, 120], evaluate=False) == [Product(i1 + 1, (i1, 1, i0 - 1))]
assert guess([1, 2, 6, 24, 120]) == [RisingFactorial(2, i0 - 1)]
assert guess([1, 2, 7, 42, 429, 7436, 218348, 10850216], niter=4) == [
2**(i0 - 1)*(Rational(27, 16))**(i0**2/2 - 3*i0/2 +
1)*Product(RisingFactorial(Rational(5, 3), i1 - 1)*RisingFactorial(Rational(7, 3), i1
- 1)/(RisingFactorial(Rational(3, 2), i1 - 1)*RisingFactorial(Rational(5, 2), i1 -
1)), (i1, 1, i0 - 1))]
assert guess([1, 0, 2]) == []
x, y = symbols('x y')
assert guess([1, 2, 6, 24, 120], variables=[x, y]) == [RisingFactorial(2, x - 1)]
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