handle ogf > 1

src/sage/combinat/cfinite_sequence.py

@@ -43,8 +43,28 @@
     sage: CFiniteSequence.guess([0,1,1,2,3,5,8])
     C-finite sequence, generated by x/(-x^2 - x + 1)
 
+TESTS::
+
+    sage: CFiniteSequence(x/(1-x))
+    C-finite sequence, generated by x/(-x + 1)
+    sage: CFiniteSequence(x^3/(1-x-x^2)).recurrence_repr()
+    'Homogenous linear recurrence with constant coefficients of degree 2: a(n+2) = 1*a(n) + 1*a(n+1), starting a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1'
+    sage: r=CFiniteSequence.from_recurrence([1,1],[0,0,0,1])
+    sage: r.seq(5)
+    [0, 0, 0, 1, 1]
+    sage: r.ogf()
+    x^3/(-x^2 - x + 1)
+    sage: r = CFiniteSequence.from_recurrence([-1],[1])
+    sage: s = CFiniteSequence.from_recurrence([-1],[1,-1])
+    sage: r == s
+    True
+    sage: CFiniteSequence(1/x)
+    Traceback (most recent call last):
+    ...
+    ValueError: Denominator divisible by x not allowed.
+
 SEEALSO:
-   :func:`fibonacci`, :class:`sage.combinat.binary_recurrence_sequences.BinaryRecurrenceSequence`
+   :func:`fibonacci`, :class:`BinaryRecurrenceSequence`
 
 AUTHORS:
 
@@ -123,14 +143,17 @@ def __init__(self, ogf):
             num = ogf.numerator()
             den = ogf.denominator()
             f = den.constant_coefficient()
+            if f == 0:
+                raise ValueError("Denominator divisible by x not allowed.")
             num = num/f
             den = den/f
             if den == 1:
                 self._deg = 0
             else:
                 self._deg = den.degree()
             R = PowerSeriesRing(Rationals(), 'x')
-            self._a = (R(num, prec=self._deg) / R(den, prec=self._deg)).padded_list()
+            alen = max(self._deg, num.degree() + 1)
+            self._a = (R(num, prec=alen) / R(den, prec=alen)).padded_list()
             self._c = [-den.list()[i] for i in range(1,self._deg+1)]
             self._ogf = num/den
         else: raise ValueError("Cannot convert to CFiniteSequence.")
@@ -168,14 +191,13 @@ def from_recurrence(cls, coefficients, values):
         if not isinstance(values, list):
             raise ValueError("Wrong type for recurrence start value list.")
         deg = len(coefficients)
-        if len(values) > deg:
-            raise ValueError("More than deg start values not implemented.")
 
         co = coefficients[::-1]
+        co.extend([0] * (len(values)-deg))
         R = PolynomialRing(Rationals(), 'x')
         x = R.gen()
         den = -1 + sum([x**(n+1)*co[n] for n in range(deg)])
-        num = -values[0] + sum([x**n * (-values[n] + sum([values[k]*co[n-1-k] for k in range(n)])) for n in range(1,deg)])
+        num = -values[0] + sum([x**n * (-values[n] + sum([values[k]*co[n-1-k] for k in range(n)])) for n in range(1,len(values))])
         return cls(num/den)
 
     def __repr__(self):
@@ -237,7 +259,7 @@ def recurrence_repr(self):
                 else:
                     cstr = cstr + ' + ' + str(self._c[i]) + '*a(n+' + str(i) + ')'
         astr = ', starting a(0) = ' + str(self._a[0])
-        for i in range(1, self._deg):
+        for i in range(1, len(self._a)):
             astr = astr + ', a(' + str(i) + ')' + ' = ' + str(self._a[i])
         return 'Homogenous linear recurrence with constant coefficients of degree ' + str(self._deg) + ': ' + cstr + astr
 
@@ -292,15 +314,16 @@ def __getitem__( self, key ) :
             m = max(key.start, key.stop)
             return [self[ii] for ii in xrange(*key.indices(m+1))]
         elif isinstance( key, (Integer, int) ) :
+            d = self._deg
             if key < 0 : #Handle negative indices
                 raise IndexError, "The index (%d) is out of range."%key
-            if self._deg == 0:
+            if d == 0:
                 if len(self._a) > key: return self._a[key]
                 else:                  return 0
-            A = Matrix(Rationals(), 1, self._deg, self._c)
-            B = Matrix.identity(Rationals(), self._deg-1)
-            C = Matrix(Rationals(), self._deg-1, 1, 0)
-            V = Matrix(Rationals(), self._deg, 1, self._a[::-1])
+            A = Matrix(Rationals(), 1, d, self._c)
+            B = Matrix.identity(Rationals(), d-1)
+            C = Matrix(Rationals(), d-1, 1, 0)
+            V = Matrix(Rationals(), d, 1, self._a[:d][::-1])
             M = Matrix.block([[A],[B,C]],subdivide=False)
             return list(M**(key-1)*V)[0][0]
         else:
@@ -357,8 +380,14 @@ def seq(self, n):
             [0, 1, 2, 3, 4]
         """
 
-        R = PowerSeriesRing(Rationals(), 'x')
-        return (R(self._ogf.numerator(), prec=n) / R(self._ogf.denominator(), prec=n)).padded_list(n)
+        if n <= len(self._a):
+            outlist = self._a[:n]
+        else:
+            outlist = self._a
+            R = PowerSeriesRing(Rationals(), 'x')
+            fullseq = (R(self._ogf.numerator(), prec=n) / R(self._ogf.denominator(), prec=n)).padded_list(n)
+            outlist.extend(fullseq[len(self._a):])
+        return outlist
 
     @staticmethod
     def guess(sequence):
@@ -404,19 +433,8 @@ def random_object():
 """
 .. TODO::
         
-        sage: CFiniteSequence(x/(1-x))         # specific output not implemented
-        Constant sequence a(n) = 1, starting a(0) = 0
-        sage: CFiniteSequence(x^3/(1-x-x^2))       # gf>1 output not implemented
-        Homogenous linear recurrence with constant coefficients of degree 2: a(n+2) = 1*a(n) + 1*a(n+1), starting a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1
-        sage: r=CFiniteSequence.from_recurrence([1,1],[0,0,0,1])   # gf>1 from attr not implemented
-        sage: r.ogf()      # not implemented
-        x^3/(-x^2 - x + 1)
         sage: r.egf()      # not implemented
         exp(2*x)
         sage: latex(r)        # not implemented
         \big\{a_{n\ge0}\big|a_{n+2}=\sum_{i=0}^{1}c_ia_{n+i}, c=\{1,1\}, a_{n<2}=\{0,0,0,1\}\big\}
-        sage: r = CFiniteSequence.from_recurrence([-1],[1])       # not implemented
-        sage: s = CFiniteSequence.from_recurrence([-1],[1,-1])    # not implemented
-        sage: r == s                                                # not implemented
-        True
         """