sagemathgh-36029: k-regular sequences: boundedness
    
<!-- ^^^^^
Please provide a concise, informative and self-explanatory title.
Don't put issue numbers in there, do this in the PR body below.
For example, instead of "Fixes sagemath#1234" use "Introduce new method to
calculate 1+1"
-->
<!-- Describe your changes here in detail -->

Implements an algorithm to check for boundedness of k-regular sequences.
See issue sagemath#22964.

(Coding and reviewing already happend on trac (see sagemath#22964) and the
branch was ready but never merged due to a trivial issue. Now the
current develop (10.1.beta8) has been merged in to avoid conflicts the
upcoming sagemath#36011 (where a renaming took place; set as depedency now).

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation accordingly.

### ⌛ Dependencies

- sagemath#36011: Experimental warning removed in class used in this PR and
class was renamed.
    
URL: sagemath#36029
Reported by: Daniel Krenn
Reviewer(s): cheuberg, Daniel Krenn, Gabriel Lipnik

Author: Release Manager

src/doc/en/reference/combinat/module_list.rst

@@ -207,6 +207,7 @@ Comprehensive Module List
     sage/combinat/ranker
     sage/combinat/recognizable_series
     sage/combinat/regular_sequence
+    sage/combinat/regular_sequence_bounded
     sage/combinat/restricted_growth
     sage/combinat/ribbon
     sage/combinat/ribbon_shaped_tableau

src/doc/en/reference/references/index.rst

@@ -5100,6 +5100,11 @@ REFERENCES:
 .. [MS1977] \F. J. MacWilliams, N. J. A. Sloane, *The Theory of Error-Correcting
             Codes*, North-Holland, Amsterdam, 1977
 
+.. [MS1977a] Arnaldo Mandel, Imre Simon,
+             *On Finite Semigroups of Matrices*,
+             Theoretical Computer Science 5 (101-111),
+             North-Holland Publishing Company
+
 .. [MS1994] \P. Martin and H. Saleur.
             *The blob algebra and the periodic Temperley-Lieb algebra*.
             Lett. Math. Phys., **30** (1994), no. 3. pp. 189-206.

src/sage/combinat/meson.build

@@ -87,6 +87,7 @@ py.install_sources(
   'ranker.py',
   'recognizable_series.py',
   'regular_sequence.py',
+  'regular_sequence_bounded.py',
   'restricted_growth.py',
   'ribbon.py',
   'ribbon_shaped_tableau.py',

src/sage/combinat/regular_sequence.py

@@ -1279,6 +1279,118 @@ def partial_sums(self, include_n=False):
 
         return result
 
+    @cached_method
+    def is_bounded(self):
+        r"""
+        Return whether this `k`-regular sequence is bounded.
+
+        EXAMPLES:
+
+        Thue--Morse Sequence::
+
+            sage: Seq2 = RegularSequenceRing(2, ZZ)
+            sage: TM = Seq2([Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [1, 0]])],
+            ....:           left=vector([1, 0]), right=vector([0, 1]))
+            sage: TM.is_bounded()
+            True
+
+        Binary Sum of Digits::
+
+            sage: SD = Seq2([Matrix([[1, 0], [0, 1]]), Matrix([[0, -1], [1, 2]])],
+            ....:           left=vector([0, 1]), right=vector([1, 0]))
+            sage: SD.is_bounded()
+            False
+
+        Sequence of All Natural Numbers::
+
+            sage: N = Seq2([Matrix([[2, 0], [2, 1]]), Matrix([[0, 1], [-2, 3]])],
+            ....:          left=vector([1, 0]), right=vector([0, 1]))
+            sage: N.is_bounded()
+            False
+
+        Indicator Function of Even Integers::
+
+            sage: E = Seq2([Matrix([[0, 1], [0, 1]]), Matrix([[0, 0], [0, 1]])],
+            ....:          left=vector([1, 0]), right=vector([1, 1]))
+            sage: E.is_bounded()
+            True
+
+        Indicator Function of Odd Integers::
+
+            sage: O = Seq2([Matrix([[0, 0], [0, 1]]), Matrix([[0, 1], [0, 1]])],
+            ....:          left=vector([1, 0]), right=vector([0, 1]))
+            sage: O.is_bounded()
+            True
+
+        Number of Odd Entries in Pascal's Triangle::
+
+            sage: U = Seq2([Matrix([[3, 0], [6, 1]]), Matrix([[0, 1], [-6, 5]])],
+            ....:          left=vector([1, 0]), right=vector([0, 1]))
+            sage: U.is_bounded()
+            False
+
+        Counting '10' in the Binary Representation::
+
+            sage: C = Seq2([Matrix([[0, 1, 0, 0], [0, 0, 0, 1],
+            ....:                   [-1, 0, 1, 1], [0, 0, 0, 1]]),
+            ....:           Matrix([[0, 0, 1, 0], [0, 1, 0, 0],
+            ....:                   [0, 0, 1, 0], [-1, 0, 1, 1]])],
+            ....:                  left=vector([1, 0, 0, 0]),
+            ....:                  right=vector([0, 0, 1, 0]))
+            sage: C.is_bounded()
+            False
+
+        Numbers Starting with '10'::
+
+            sage: D = Seq2([Matrix([[0, 1, 0, 0], [0, 0, 1, 0],
+            ....:                   [0, -2, 3, 0], [0, -2, 2, 1]]),
+            ....:           Matrix([[2, 0, 0, 0], [0, 0, 0, 1],
+            ....:                   [0, 2, 0, 1], [0, -2, 0, 3]])],
+            ....:                  left=vector([1, 0, 0, 0]),
+            ....:                  right=vector([2, 2, 2, 5]))
+            sage: D.is_bounded()
+            False
+
+        Signum Function::
+
+            sage: S = Seq2([Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 1]])],
+            ....:          left=vector([1, 0]), right=vector([0, 1]))
+            sage: S.is_bounded()
+            True
+
+        Number of Digits from the Right to the First '1'::
+
+            sage: S = Seq2([Matrix([[0, 1, 0], [-1, 2, 0], [0, 0, 1]]),
+            ....:           Matrix([[0, 0, 1], [0, 0, 2], [0, 0, 1]])],
+            ....:          left=vector([1, 0, 0]), right=vector([0, 0, 1]))
+            sage: S.is_bounded()
+            False
+
+        .. SEEALSO::
+
+            :mod:`boundedness of k-regular sequences <sage.combinat.regular_sequence_bounded>`
+
+        TESTS::
+
+            sage: S = Seq2((Matrix([[0, 1, 0], [0, 0, 1], [-1, 2, 0]]),
+            ....:           Matrix([[-1, 0, 0], [-3/4, -1/4, 3/4], [-1/4, 1/4, -3/4]])),
+            ....:          left=vector([1, 0, 0]), right=vector([-4, -4, -4]))
+            sage: S.is_bounded()
+            False
+
+        ::
+
+            sage: S = Seq2((Matrix([[1, 0], [1, 0]]), Matrix([[0, 1],[1, 0]])),
+            ....:          left = vector([1, 1]), right = vector([1, 0]),
+            ....:          allow_degenerated_sequence=True)
+            sage: S.is_degenerated()
+            True
+            sage: S.is_bounded()
+            True
+        """
+        from sage.combinat.regular_sequence_bounded import regular_sequence_is_bounded
+        return regular_sequence_is_bounded(self)
+
 
 def _pickle_RegularSequenceRing(k, coefficients, category):
     r"""